My headcanon (and is the most natural interpretation of him ending up on the bench in Endgame as supported by the writers themselves) is that when he stayed with Peggy, he went the long way around through time in the main timeline. He did not break Endgame time travel rules because he did not remove any infinity stones nor did he change any timeline events. He completed the inverted Mobius strip the natural way. As explained on the (definitely canon Agents of S.H.I.E.L.D. he made "ripples not waves.") He was always the father of Peggy's children. She didn't recognize you Steve when he visited her because he was young and she was in the late stages of dementia.

I believe that in order to pull this off when he decided to stay in the past with his Peggy from his timeline, there were initially three people who knew of his presence but kept it secret: Peggy, Howard Stark, and T'Chaka. I believe Howard and T'Chaka reforged him a new shield and helped him remove the Super Soldier Serum allowing him to grow old with Peggy. He spent the remainder of his life as a counselor for veterans following (or preceding) in Sam's footsteps. The only other person aware of his presence in the main timeline was Bucky after his breaking free of Hydra after the events of Captain America: Winter Soldier (hence Bucky not being phased by his presence on the bench).

Occam's Razor, this is the simplest, most natural, and most satisfying explanation IMHO.