Your argument is faulty because you make a wrong assumption. You are arguing that Steve staying in the past altered the past, when he in fact always has been part of the past.
Steve's life plays out like this:
He becomes the super soldier, is frozen in ice for 70 years. He wakes up in 2012 and fights with the Avengers. After beating Thanos in 2023, he travels back in time to put the stones back. After completing this task, he travels back to after he first got put on ice, "retires" as a superhero and starts living with Peggy. He grows old and in 2023 goes to the place his young version travels back, to hand the shield to Sam and make sure the future will still have a Captain America.
This has always been the chain of events, and the moment Steve decides to stay in the past is merely the "entry point" of the loop from our perspective. But during the events of Iron Man all the way up to that entry point, old Steve has been there.
The only thing they could've done to make this more clear is show extensive scenes of old Steve witnessing the events of the MCU but knowing not to step in, showing scenes of him living out a normal life under a different identity and Peggy keeping their life as much a secret as possible as well as old Steve saying goodbye to Peggy before she dies and watching her funeral in secret to tie up some loose plot ends. But all this is really stuff you should figure out yourself based on the theoretical rules of time travel.