I disagree, I don’t think it broke anything, but as with any case where time travel makes an appearance, it can get convoluted really quickly. I kinda want to work it out in writing if only for my own sake, but I’ll put it in a cut since it’ll be rambling and loaded with spoilers…
SPOILERS and timey wimey nonsense rambling, you've been warned...
First off, the movie actively poo-poos time travel as conceptualized in modern culture (“Man, Back to the Future was bullshit!”), which is probably the biggest ask in the entire film. The more I think about it, the wilder it is that the Russo brothers built the film’s entire narrative around a completely different model of time travel and expected the audience to latch onto it in less than 3 hours.
Essentially, it’s impossible to create a time paradox in the MCU, because any changes you make simply branch the timeline into a new reality. Where you are in time is always “your” present, and regardless of when in time something happened, it always happened in “your” past. (I believe Hulk’s line was something along the lines of “If you go back in time, your past is now your present and your future is now your past.”) Anything you’ve already experienced remains your reality, so the snap remains, in the parlance of Doctor Who, a fixed point in time for our heroes (emphasis to become important later). It can’t be undone, but it can be worked around. The other difference, from what I can see, is that unlike Back to the Future, if you change something in the past, you don’t travel forward to that reality’s future, you just go back to your own. People are inexorably tied to their own reality, because quantum mechanics can basically do whatever you really need it to do.
Removing the infinity stones from the existing timeline therefore doesn’t create any risks to the timeline as we’ve experienced it in the films, it simply creates new branches we don’t get to see, where the stones vanish. At the insistence of the Sorcerer Supreme, the stones have to be returned to close off those realities and prevent them from occurring once the Avengers are done un-breaking their own universe, because apparently Bad Things happen in them—the Time Stone-less reality in particular. As far as I can tell, that puts us in a universe where, at the end of Endgame, the stones are still destroyed in our timeline. I’m sure they could find a way to bring them back, but it would involve a considerable amount of effort and a lifetime supply of handwavium.
The biggest hitch in the timeline is that Thanos comes to 2023 and gets himself snapped, thus preventing his re-insertion back into the timeline of 2012 onward. However, that doesn’t really break the film continuity, because as previously explained, it simply creates a new branch of reality where Thanos vanishes before he ever gets his hands on any of the stones. What we experienced in the MCU is the timeline where that didn’t happen, so that it could happen in Endgame. Presumably, the Sorcerer Supreme is fine with leaving that reality un-aborted because now that reality doesn’t have to deal with a megagenocidal superlunatic anymore. I was actually expecting the movie to put Thanos back, so he’d return to the past with the knowledge of where the stones were and thus be able to acquire them for his gauntlet in the first place, but then I realized that wouldn’t really make sense because he’d also know he had to nuke-and-pave the whole cosmos from the get-go to prevent the Avengers from messing up his whole… thing.
The only unresolved time travel goof is when Loki takes the Space Stone after Stark gets knocked out by the Hulk coming down the stairs. And again, that doesn’t break the continuity of The Avengers, because The Avengers as we saw it occurred in the past from our heroes’ perspective, so Loki fucking off to wherever is simply a branch that doesn’t get closed off again the way the other 6 do. Presumably, this is the setup for the Loki spin-off TV show being made for Disney+, because Loki died in the first 15 minutes of Infinity War and is otherwise accounted for in the rest of the MCU’s timeline.
Rogers can still go back to be with Carter and show up again at the end because he arrives to be with her before any of the other timeline shenanigans occur, so he’s not in one of those alternate branches of reality, and there’s plenty of time after Carter’s TV show for him to pop back up without doing anything but recast Carter’s presumed pining in various flashbacks (it would also explain why they stopped looking for the gigantic black plane that crashed in a freaking snow field). I’d also have to re-watch Winter Soldier to see how this slots in with Steve’s reunion with Carter right before she dies. It might screw that one up, but I kind of don’t care because America’s ass deserves to be happy god dammit!
All of that said, I do agree it’s not the best Marvel film… it’s big and bulky and has a lot to do, and even in 3 hours it has to race through a lot to do it. It takes something of a Thor-wielding-Mjolnir approach to getting the job done, and some of the character beats fall really flat (like everyone constantly ribbing on Thor, who quite clearly has PTSD in the wake of Infinity War and the whole photograph bit in the diner with Hulk and Ant-Man). I still think it’s impressive that they managed to build something this crazy-complicated and interwoven with any amount of competency whatsoever. It’s just so many moving parts and juggling balls to keep track of at the studio level.