I'm just focusing on one particular scene here, wherein Doug Carlin, played by Denzel Washington, throws a kind of tantrum in a control room, breaking a monitor.
Why does he do this?
He's being let in, as an insider, to the core counterterrorism center, the better to do his job. The monitors show events four days in the past, which the experts claim is based on satellite feed and some post processing.
Carlin does some empirical testing of his own and confirms his suspicion that he's being lied to, which is when he throws his fit. How can he really do his job most effectively if he doesn't have a realistic grasp on what his degrees of freedom really might be? Is he actually able to retroject himself into the past? He really needs to know this, and he tries to hammer that point home.
I found this scene effective in that I wasn't buying the satellite feed model either, as I kept telling Tara. Besides, we both knew from the previews that time travel was somehow involved. My impatience found cathartic expression in Carlin's throwing a fit. He's just trying to do his job, so why all the BS?
Carlin and the other controllers have witnessed a horror and will do pretty much anything to prevent it from happening again -- or might they really undo the past itself, even at the cost of erasing the present? Yes, apparently.
Within the domain of timeloop science fiction (12 Monkeys being another of that genre), this one resolves nicely, like a clever slip knot.