I believe the first part of your question (your assumption regarding the algorithm being delivered into the future) is correct. The answer saying otherwise is unsourced; let's explore an alternative approach.
A Primer on Information (And Who Has It)
It is important to recognize throughout the film that most characters, The Protagonist (TP) included, do not posses all, most, or even a little bit of the information required to discern the nature of the plan enacted by Tenet beyond what is required for them to perform the actions or deliver the information for which they are personally responsible. This theme is highlighted in the following lines of dialogue:
FAY
There's a cold war. Cold as ice. To even know its true nature is to lose.
NEIL
The policy is to suppress.
IVES
Need to know. And you don't. Any other stupid questions?
This theme will guide us in the assumption that, unless otherwise noted, characters posses no information relevant to Tenet/the plan other than what has been specifically provided to them. This is made explicitly clear by Fay, at the close of their conversation on the deck of the ship:
FAY
This is knowledge divided – all I
have for you is a gesture –
(interlaces his fingers)
– in combination with a word –
‘tenet’. Use it carefully, it’ll
open the right doors, but some of
the wrong ones, too.
PROTAGONIST
That’s all they’ve told you?
FAY
The test you passed?
(beat)
Not everybody does.
These lines establish Fay as a pawn, given one role (transmit the two items of information) in accordance with his (lack of) ability--an implied unwillingness to commit suicide instead of divulging important information. This is further shown on a procedural/organizational level during TP's conversation with the scientist after his departure from the windmill ("no small talk").
We will assume that the ultimate source of the information disseminated by Tenet to its members is TP, in a plan enacted after* the events of the film. This is based on the conversation TP has with Neil prior to Neil and Ives departing from the hypocentre LZ. We will assume, based on the length and nature of their relationship (and narratively from Neil's role as a source of reliable exposition), that Neil has adequate information to accurately conclude that TP is indeed the founder. This is stated clearly at the end of the film, but the film is convoluted enough that nailing down the obvious is a good first step.
*From TP's perspective
Good on you if you've made it this far. So why is this extreme level of information compartmentalization necessary?
The Nature of the Conflict
The cold war referenced by Fay is an existential conflict between The Present and The Future, with The Future desiring to reverse the ecological destruction of the earth. From the conversation between TP and Sator by sat-phone in the hypocentre:
PROTAGONIST
How can they want to destroy us?
SATOR
(over phone)
Because their oceans rose and their
rivers ran dry. Don’t you see –?
Their world shrivelled because of
us. They have no choice but to turn
back, there’s no life ahead of
them. And we’re responsible.
Knowing this, do you still want me
to stop?
PROTAGONIST
Yes. Each generation looks out for
its own survival.
The two sides are locked in an annihilatory struggle (annihilate--"to move to nothing"--seems especially appropriate in this case). The Present is threatening the existence of the The Future through wanton ecological destruction. The Future, through implementing the algorithm, is threatening the existence of The Past.
The Nature of the Algorithm
PROTAGONIST
What’s the ‘algorithm’, Neil?
NEIL
The 241 is one section of it. One
out of nine. It’s a formula
rendered into physical form so it
can’t be copied or communicated. A
black box with one function.
...that function being...
NEIL
Inversion. But not objects or
people. The world around us.
... which is necessary because, in a conflict involving objects sent through time, the default direction of entropy favors The Present:
NEIL
Because the environment’s entropy
flows in our direction we dominate.
They’re always swimming upstream.
"Swimming upstream" is not rigorously defined in the logic of the movie ("don’t try flying a plane – it’d fall out of the sky"--a line that didn't make the final cut--suggests that the more complex the interaction the more resistance there is), but we'll assume that it's the dynamic responsible for TP's knife wound appearing as he travels back in time/the implied gradual appearance of the bullet holes in the glass in the Oslo vault (as opposed to them always being there).
The actual Cold War (ours, First vs. Second world, capitalism vs. communism) offers a potential explanatory metaphor: defection. If one side offers a higher quality of life individuals will naturally gravitate towards it, tangent to any ideological parameters. We can apply the metaphor to physical objects in a similar manner; they naturally gravitate towards the future, with objects travelling in the opposite direction inclined to "defect"/degrade/disappear. This metaphor is entirely my own attempt at an explanation, and can be disregarded at will.
It seems dicey to introduce the term "break causality" into such an already rickety structure, but we can speculate that reversing the flow of entropy would cause the present to pass out of existence much like TP's knife wound came into it; viewer speculation aside, we have the speculation of Neil in the film to go on, with the ultimate result of implementing the algorithm uncertain:
PROTAGONIST
I’ve been thinking... we’re their
ancestors. If they destroy us,
won’t that destroy them?
NEIL
Bringing us to the grandfather
paradox.
PROTAGONIST
The what?
NEIL
The classic thought experiment – if
you went back in time and killed
your own grandfather, how could you
have been born to commit the act?
PROTAGONIST
What’s the answer?
NEIL
There is no answer, it’s a paradox.
But in the future, those in power
clearly believe that you can kick
Grandpa down the stairs, gouge his
eyes and slit his throat without
consequence.
PROTAGONIST
Could they be right?
NEIL
It doesn’t matter. They believe it,
so they’re willing to destroy us.
Can I go back to sleep?
The explicit message of this conversation is that the effects of implementing the algorithm are A. uncertain but B. almost certain to result in the destruction of The Present. Whether that would result in the destruction of The Future is stated to not be clear, but it's indicated that the residents of The Future are desperate enough that they're willing to take the chance and "kill Grandpa". The scientist responsible for inventing the algorithm disagreed:
PRIYA
Think of our scientist as her
generation’s Oppenheimer – she
devises a method for inverting the
world, but becomes convinced that
by destroying us, they destroy
themselves.
...and so sends it into the past, as Oppenheimer might have taken steps to destroy the Manhattan project had he believed that detonating a nuclear weapon would ignite the atmosphere. (There's an interesting question there: if the scientist had been confident that The Future would survive the destruction of The Present would she consent to the activation of the weapon, like Oppenheimer did?)
The Functioning of the Algorithm
This brings us to the nuts and bolts of the functioning of the Algorithm, specifically as they relate to the events at the hypocentre. There are a few key points:
It is not indicated that the detonation of the bomb at the top of the hypocentre shaft is responsible for the activation of the algorithm. From the conversation on the ship about the dead man's switch while TP, Neil, and Kat are spending a week inverting for the final operation:
NEIL
It’ll be linked to a switch.
Probably a simple email burst,
revealing the location of the dead
drop, set to fire if his heart
stops.
PROTAGONIST
In effect, his death activates the
algorithm. He dies, the world ends
– no one dares kill him.
The operative term in this case being in effect. Sato's death/the sending of the dead-man's switch e-mail is not directly related to the explosion that happens at the top of the shaft, but is instead a fulfillment of his mandate from the future:
PRIYA
Sator’s lifelong mission, financed
and guided by the future, has been
to find and reassemble the
algorithm.
PROTAGONIST
Why did they choose him?
PRIYA
The necessary combination of greed
and ambition. But mostly, he was in
the right place at the right time.
...to assemble the pieces of the algorithm and place them in a location where they can later be found, supported by the earlier establishment of the concept of the dead drop:
PROTAGONIST
Dead drops. He buries his time
capsule, transmits the location,
then digs it up to collect whatever
inverted material they’ve sent.
...just in the case of the algorithm, instead of The Future placing inverted objects in the capsule to be dug up in the past, Sator is placing an un-inverted object to be found at a later date (a traditional time capsule).
It is indicated that the algorithm will be activated in the future. From the sat-phone conversation with Sator in the hypocentre:
PROTAGONIST
(over phone)
What’s more fanatical than trying
to destroy the world?
SATOR
I’m not. I’m creating a new one.
Somewhere, sometime, a man in a
crystalline tower throws a switch
and Armageddon is both triggered
and avoided. Entropy inverts the
same way the magnetic poles have
switched 183 times over the
millennia. Now time itself switches
direction.
The "man in the crystalline tower" is an individual in The Future ("sometime") activating the implementation of the algorithm, inverting entropy. This avoids the ecological Armageddon of future earth, and triggers the temporal Armageddon of The Present.
The assumption that the method to implement the algorithm is to destroy the hardware on which it is implemented does not make a ton of sense. Additionally, the algorithm is specifically said to be sealed within the hypocentre during the conversation in which TP reveals the location of the dead drop to Ives:
PROTAGONIST
Ground zero for an underground
nuclear test. Sir Michael Crosby
told me about a detonation at
Stalsk-12 on the 14th – the dead
drop is at the bottom of the
hypocentre. That explosion seals up
the algorithm.
IVES
Then we’d better pull it out of
that hole before the bomb goes off.
Reinforced by the line from the script stating Volkov (final henchman/knuckle tattoo enthusiast/anti-hot sauce mastermind) is struggling for the winch release--if the purpose was to detonate a bomb next to the algorithm, what purpose is served by dropping it down a hole, away from the explosion?
INT. TUNNEL, STALSK-12 – CONTINUOUS
The Protagonist DIVES to where Ives is lying, GRABS his gun –
Volkov TURNS, RUNNING FOR the winch-release –
There is none.
Answering Your Question
At this point things get a little hazy. We've established that the movie took a pass on explaining the grandfather paradox; it's attempt at nailing down the nature of causality can be summed up in this exchange:
NEIL
What’s happened’s happened.
(smiles)
Which is an expression of faith in
the mechanics of the world, not an
excuse to do nothing.
PROTAGONIST
Fate?
NEIL
Call it what you want.
PROTAGONIST
What do you call it?
NEIL
Reality. Now let me go.
We assume that, given time and excavation equipment, TP/Tenet would have had no problem retrieving the algorithm from the hypocentre. But that assumption has a Present bias; based on the logic presented in the movie it would be equally valid to assume that, had the algorithm been successfully sealed in, The Future would have had no problem retrieving it and, from the perspective of The Present, immediately erasing the existence of the latter. This did not occur; what happened happened, and reality as the characters know it was a product of their actions.
That The Present continued to exist suggests that The Present/Tenet was "always" going to be successful in stopping the transfer of the algorithm to The Future. This requires viewing events in the future as functionally the same as events that have occurred: that they always were/always am, and that reality is constructed simultaneously instead of sequentially (tough to wrap your mind around when we obviously perceive it very differently).
You could imagine that someone struck and killed by a car could have been saved by standing three feet to the left, or you could imagine a car that swerves three feet to the right could strike and kill someone who would have otherwise been unharmed, but the individual is either struck by the car or they are not struck by the car. Consider the meta-physical car (Sarto assembling the algorithm and broadcasting its location) sent by The Future to kill The Present: whether it succeeds or fails, the fact remains that it can only be one possibility or the other. The turnstile technology, by turning time into a dimension that can be traversed like any other, and therefore removing the independent variable by which we usually define events as exclusive to each other, flattens our perception of what is and isn't possible--everything "just is", "always was", and "always will be", which would be the mechanics of the world that Neil expresses faith in*.
*Comments on nature of reality not substantiated, safety during time travel not guaranteed.
This interpretation is supported by the role of information in the film (coming full circle back to the importance of information security). During TP's first conversation with Priya:
PROTAGONIST
He can communicate with the future?
PRIYA
We all do. Credit cards, email,
text – anything that goes into the
record speaks directly to the
future. The question is – can the
future speak back? And, if so, what
are they saying?
The "always is, always was" nature of communication (where anyone can communicate into the future by making a record of it) means that nothing is ever "over", only lost and forgotten--The Future only has to "get lucky" once, while Tenet must be successful forever (but that The Present continues to exist implies that they are).
