Sator has a dead man's switch which will detonate the algorithm and reverse entropy, ending the world.
Sator's dead man's switch by itself will not detonate the algorithm. It just sends out the location of the dead drop (the buried capsule). The people in the future (posterity) are expecting Sator to send them the location of the assembled algorithm (inside the capsule / dead drop), so they could get it in the future and trigger it then.
Since the Tenet team was able to lift the algorithm from the dead drop location, even if the dead drop location has been sent to posterity, it can not be triggered (because it's not there anymore).
Related dialogue (emphasis mine):
On communicating with the future/posterity
Protagonist: [Talking about Sator] He can communicate with the future?
Priya: We all do, don't we? E-mails, credit cards, texts. Anything that goes into the record speaks directly to the future.
Protagonist: Dead drops.
Protagonist: He buries his time capsule, transmits the location, then digs it up to collect the inverted materials they sent.
Neil: Seemingly instantaneous.
On who will trigger the algorithm
Priya: Sator's lifelong mission, financed and guided by the future, has been to find and reassemble the algorithm.
The people in the future financed and guided Sator to find and reassemble the algorithm, not to trigger or activate it.
Protagonist: The fitness tracker he wears.
Neil: It'll be linked
to a switch, probably a simple e-mail burst that reveals the
location of the dead drop, set to fire if his heart stops.
The fitness tracker's dead man's switch only reveals the location of the dead drop to posterity. If Sator was meant to trigger or activate the algorithm himself, in his present, there would be no need for him to reveal the location of the dead drop to posterity.
Protagonist: [To Kat] You're not there to kill him, you're the backstop.
Protagonist: If we don't lift that algorithm and he kills himself, he takes us all with him.
Note the words: "backstop" and "and". Kat preventing Sator from killing himself is just the "backstop" and not the main part of the operation. The main part of the operation is the temporal pincer to lift the algorithm from the dead drop location. Kat failed to prevent Sator from dying (and actually caused his death herself), but the algorithm didn't trigger or activate. If Sator's dead man's switch was meant to trigger the algorithm, then it's "end of play"—to use Neil's words—as soon as Sator got killed. The Tenet team would not be able to lift the algorithm from the dead drop location, because the algorithm was still inside the cavern / dead drop location when Kat killed Sator.
Protagonist: What's more fanatical than trying to destroy the world?
Sator: I'm not, I'm creating a new one.
Sator: Somewhere, sometime, a man in a crystalline tower throws a switch and armageddon is both triggered and avoided.
Sator says that he is not destroying the world, but "creating a new one" (for the people in the future). Sator did not refer to himself or his present when talking about who will trigger or when armageddon (the algorithm) is triggered.
Sator: Now time itself switches direction. The same sunshine we basked in will warm the faces of our descendants' generations to come.
Protagonist: How could they wanna kill us?
Sator: Because their oceans rose and their rivers ran dry. Don't you see?
They have no choice but to turn back. We're responsible.
Sator: Knowing this, do you still want me to stop?
Protagonist: Yes. Each generation looks out for its own survival.
Sator: That's exactly what they're doing.
Protagonist: But not you. You're a traitor. Bringing death to all, because
you have no life of your own left.
Sator: When I'm done, life continues.
The main goal of the people in the future is not to destroy the world. Their main goal is their survival, to save their own world, whose environment is so degraded that it can barely support life. Sator (who is financed and guided by the future) detonating the algorithm in his present does not make sense if the main goal of the people in the future is to save their own world.
Protagonist: I'd like to say... That you don't have to do this, Kat...
Kat: Worst thing Andrei ever did to me was that offer he made me.
Kat: Let me go if I agreed never to see my son again.
Kat: A chance to help save my child. You can't know what that means to a mother.
Mahir: Ives, she's killed him. Ives, do you copy? She's killed him.
Ives: She jumped the gun. She killed him.
Protagonist: Kat! You jumped the gun!
Kat: I couldn't do it. I couldn't let him die thinking he'd won. I knew you'd find a way.*
Kat: Wait, you found a way, we're okay, right?
Protagonist: Yeah, found a way*. Be safe.
* lift the algorithm from the dead drop location; also, refer to the "backstop" dialogue
Kat killed Sator knowing that it will just send the location of the algorithm to posterity, and not that it will trigger armageddon. Throughout the film, it is shown that Kat cares very much for her son, Max. It is the driving force for Kat's actions throughout the film. She was deeply hurt when Sator coerced her not to see Max again.
It would be against her character and previous actions in the film if she killed Sator under the assumption that his death will trigger the algorithm, destroying everyone in the present, including her son.
She "knew" that the Protagonist and team would still be able to find a way to lift the algorithm from the dead drop location.
Overall, there is no line of dialogue in the film that suggests that Sator is to be the one that detonates the algorithm. All relevant dialogue and character actions say otherwise. It is not sure if Sator even knows how to trigger or detonate the algorithm.