This scene specifically says the timelines collapse and that it is the infinity stones that control the flow of time.
"If I give up the Time Stone to help your reality, I'm dooming my own... In this new branch reality, without our chief weapon against the force of darkness, our world would be overrun. Millions would suffer."
"Once we're done with the Stones, we can return each one to its own timeline at the moment it was taken. So chronologically,
in that reality, it never left."
"I can't risk this reality on a promise."
None of that says the timelines collapse. It says the exact opposite -- that it's the continued existence of the timeline that has to be protected by returning the Stone to the moment it left so that it's like it's never gone. The whole point of their conversation is that returning the Stone will
preserve the branch reality the Ancient One occupies, not that it will erase it. She doesn't
want it erased or otherwise endangered, so Bruce promises to return the stone so that the separate reality will still exist safely.
Yes, Bruce says "we can erase it," but that contradicts what he says in the rest of his explanation, so it's just a poor choice of words. He just means they'll pre-emptively prevent the hypothetical doom that would happen in that reality if they didn't return the Stone. There's nothing to erase, because they return the Stone before it happens.
Even if there are several versions of the timeline, they all flow in the same direction.
Well, yes, that's the whole point, that alternate timelines coexist in parallel rather than "overwriting" each other. That's proven by the fact that taking the past Thanos and Gamora out of time doesn't erase what Thanos did in the original timeline; it just creates a new version of Thanos and Gamora that coexists with the original history. The plot of the movie absolutely
depends on timelines coexisting as in real scientific theory, rather than replacing each other as in fantasy.
So there is nothing to say on screen that the Rogers we see at the end of the movie hadn't just lived his entire life anonymously until he met up with Sam.
Except that's horrifically out of character for Steve Rogers. It's as impossible for him to stand by and let an injustice go unconfronted as it is for a timeline to be erased. And the logic of the movie is that just going back in time creates an alternate timeline that you occupy from that point forward; therefore Steve was in an alternate past, and at some point he managed to cross from it into the so-called "Sacred Timeline" to wrap up the movie (perhaps in the same way that the Agents of SHIELD were able to cross back from the alternate timeline where they ended up in the final season).
Egad, why does anyone think motion smoothing looks good?