There are two different types of time travel situations in Endgame: the Time Heist revolves around *removing Infinity Stones from the timeline*, Cap's journey back to Peggy (as well as Hawkeye's test run) is just normal time travel.
The ancient one's branching explanation *explicitly* only applies to removing Infinity stones. That is why the deaths of Thanos and Nebula et al mean nothing, because they all came from a branching timeline that was created by the removal of the power stone from their original timeline. Their deaths would have been noted in their own timeline if Cap hadn't subsequently erased it by returning the power stone, but he did, so none of them ever existed in the first place (except during the brief window of time when they did exist - Time Travel!). None of this can really be disputed based on the actual text of the film.
No explanation is ever given for whether 'regular time travel' creates alternate universes or predestination paradoxes or both/either depending on the circumstances. All of Hulk's scenes are so heavily laymanized that it's impossible to derive any clear rule from anything he says other than the most simplified version: 'you can't change your past', which fits either model. One simply can't rule out the possibility that there was always a second Hulk in New York talking to the Ancient one and that Hawkeye and Black Widow were always on Vormir together and that Thor always had a secret meeting with Frigga and Steve and Tony always wandered around that SHIELD base.
But there are a few different questionable parts to help you choose which explanation you like better:
Loki's escape can't be entirely ruled out as having always happened yet, but it almost certainly will be ruled out when the Loki series is released, and he also can't be easily explained by the infinity stones because no one had taken a stone from New York yet at that point. This is the primary reason I expect the alternate timeline theory will ultimately become undeniable as the mcu moves forward, even though I prefer the fixed timeline theory.
Also, while the events surrounding GotG can't be 100% ruled out as having happened (Star-Lord getting knocked out, Nebula getting weird interference and Thanos seeing some of future nebula's memories), they do seem extremely unlikely. All of this happened, again, before the power stone was removed.
On the other hand, the movie hammers hard on the absolute necessity to 'trim the branches' by returning all the stones after they're done. If alternate timelines existing is totally normal/no big deal, then this aspect of the film makes no sense. Why knowingly return the scepter and the tesseract to Hydra's hands or the Aether to Malekith if you don't actually have to? Why bother to return any of them, really (except the Time Stone, which the Ancient One explicitly wanted back)? The movie never provides any plausible reason except to 'trim the branches'.
There are various fictional cheats which can be imagined - maybe removing any stone anywhere in time ripples all the way forward down the timeline, so Loki and the Guardians were all affected by the removal of the tesseract from the SHIELD base and returning that stone undid literally every other aberration we saw in the film. Maybe removing a stone is so powerful that it screws with reality even to a limited extant before the stone is removed, which could also cover Loki and the Guardians. Or maybe alternate timelines are really normal, but Infinity stone alt. timelines are special and bad and so must be undone, thereby explaining why the need to trim the branches.
For me personally, I choose fixed timeline, because I think Steve's ending is much better if he's always been there than if he goes off to a different universe and because I think it's much more elegant for normal time travel to be explicitly different from the infinity stone alternate universes rather than they all create alternate timelines but some of them are bad. But as I said, while I 100% consider this the best, most logical interpretation of Endgame specifically, I don't think that explanation will remain consistent with where the MCU goes moving forward (through Dr Strange's Multiverse of Madness and Loki's D+ series), so I'll likely have to let go of it eventually.
As for Cap's predestination paradox being out of character, I don't buy that for a second. I think it's entirely believable for him to understand that trying to change certain things could cause something far worse, but - if you truly can't change your past - it wouldn't matter whether he ever gave into temptation or not. He could've tried to save Bucky a dozen times and just never succeeded. He could've fought an entire covert war against Hydra with Peggy's help and the best the two of them could achieve was to keep Hydra from taking over even earlier than TWS. He also could've spent an entire career, in the shadows likely under an assumed name, saving all sorts of people from all sorts of dangers that the audience never knew existed before. There is no reason why Cap going to live in his own past need result in him just permanently retiring and living a normal life for decades.
