I do not think Captain America lives out his life in our reality. Similar to how the stones were taken in a different reality he just went there and messed further with their timeline.

Here are the reasons I think this must be true.

 1. Anything else would go against all the rules of time-travel established in the movie
 2. Captain America knows Becky is alive and SHIELD is interrelated by Hydra. I am sure that in his new reality he fixes both of these things, but not in ours.

Afterwards he comes back to us by some means. Maybe he gets more Pim [particles from his version of Dr Pim at some point. Maybe the Sorcerer Supreme helps him. He meets her to give back the Time Stone.

The only question is why does he not come back to the platform? That must be just for dramatic effect. He had a lifetime to plan this moment.

EDIT:

Before anyone says that Cap would not mess with the timeline, if he did go back to our time, he clearly has shown that he does not care about it. If he risked changing something just for himself, he defiantly would have changed things to stop his best friend from being tortured.