Time Travel and Caps SL

The film made it clear that their version of time travel was far different from Back to the Future and others and it was made clear several times. Per BTTF seeing your other self could create a paradox and we see that never happened when 2023 Cap saw his past 2012 self. Same with Nebula. Per BTTF, she she should have been erased from existence once she shot her past self. But she wasn't.

As for Caps ending, I'm assuming from watching the movie that Old Cap was sitting on that bench all along(before Young Cap went back) and was only noticed when young Cap didn't show up. Also I see alot are salty because he passed it to Sam instead of Bucky. While Bucky was his close friend and probably to some was the obvious choice, it made more sense for him to pass it Sam. Remember Bucky was missing for years and was thought to be dead. So those years he was missing, Cap formed an even greater bond with Sam. I think even Bucky knew Sam was the obvious choice too and that's why he nodded for him to go talk to Old Cap sitting on the bench.