I always saw Twelve Monkeys as being centered around the fact that the past cannot be changed. But why is that? Is it a physical law that makes it impossible to change or is it just the scientists that don't want to change it for any reason?
Caveat: Looking for logic in time-travel movies is an exercise in recursive futility.
The idea in 12 Monkeys is that the past is, in many ways, read only. You can go back and "read" all you want, but you won't be able to "write" anything that will (meaningfully) change the timeline. Alternatively, you can attempt to change an event, but since the timeline is already set, it accounts for your "hack" and adjusts events to reconcile the "new" timeline to the original version.
Keeping the above in mind, the scientists in the future are content to send people into the past to get information and communicate it back to them in the future. The scientists are aware of the paradox that if they could change such a catastrophic event in the past, then they wouldn't be considering doing so at all (in their present). Presumably they have tried to change things in the past and know that it's something impossible to do. They are, therefore, happy to use time travel purely as an archaeological avenue to find out enough information about the virus to allow them to synthesise a cure in their present.
This subject is tackled to an extent in an IMDb FAQ:
There are two approaches to time travel science fiction:
In 12 Monkeys, no matter what the actions are, the consequences are always the same. E.g. Red Army herring, Releasing the virus, Cole appearing in various books. So it is more like the first time function. Cole may go in his past, but in no way shall his actions change the present. What that can be interpreted as is that what has happened, will happen. No matter what actions you might try to take to prevent it. His being shot in front of his younger self is explainable as it is something the adult Cole has lived with his entire life. He remembers the Woman's eyes in his adulthood even though he was a kid and saw her just once. Thus, these events has already happened in Cole's past. And hence, it was impossible to change that past. This is also why Cole was sent on a recon mission, to observe and to report.
Also, I would like to point out to you that Cole was given the gun at the airport. At that point of time, Cole had defected and could not be contacted. So why did they give him a gun? They had other people in the past who could have stopped the guy just as well... Just a point to ponder upon. Had Cole not tried to shoot the guy, would his younger self have seen the shooting and the Young Woman? That's the time-paradox, which cannot happen at all in such time lines. Hence the linear consequence of the timeline and the unchanging past.
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From what I gathered:
Our protagonist was unable to change the past because it was part of his present. He could have changed everything had he been told by someone else sent to him. This person would have to have gone to a timeline where the same big event ocurs, found out the truth and then go back to our progagonists timeline to let him know
The people in charge could have changed things (I think) but didn't as it's likely they wouldn't be in charge in the altered reality. They were in a position of power that was too beneficial to them to care about everyone else.