Let me know if there's any flaws in my reasoning, or events in the film that I overlooked that contradict my reasoning, but here's a couple of points I gathered:
- The wrist-mounted devices appear to be able to shift a person backwards to an earlier date, and also, it can move the wearer "sideways" into a different alternate timeline.
- But going forwards appears to require an "anchor", in the form of the time travel platform.
I came to the first conclusion because at the end of the movie, they used the platform to zap Captain America back in time with all the Infinity Stones and Mjolnir, and apparently expected him to be able to replace all those objects in all the many separate timelines without needing to repeatedly return back to the platform to get zapped to the next destination.
Therefore, the wrist devices must, at the very least, be able to send a person not just to an earlier point in their own timeline, but also to coordinates in completely separate branches (all the branches created by taking the Stones earlier).
The second conclusion is not certain, but every instance we've seen of them traveling forwards in time (this is invariably a "return trip" back to where they departed) always ends with them landing directly on an active platform. This includes Thanos' ship moving forwards in time while also skipping timelines, to arrive at the platform that was switched on by Bad Nebula.
Thus, it seems possible that jumping forwards requires an activated "listening" platform to be waiting there. Maybe this is because there are many (infinite?) possible futures, so even the insanely capable wrist devices need to know which future to land in.
So my take on Old Man Rogers is this:
After leaving the platform in 2023, he jumped between various timelines where they had earlier taken Infinity Stones (and Mjolnir), replacing them along the way. He does this in order going back and back further in the past (each hop is between different timelines as well). I suppose this means 2014 Morag (Power Stone and Soul Stone), 2013 Asgard (Reality Stone and Mjolnir), 2012 New York (Time Stone and Mind Stone), 1970 SHIELD base (Space stone), in that order.
Then he went back to the late 1940s after WW2 to look for Peggy.
Since the watch can only send a person backwards in time (even if they skip between existing timelines), he must have grown old in that separate timeline, until 2023+ at a calendar date later than his departure from the time travel platform. Then the watch allows him to travel backwards from the Peggy timeline, while also shifting to the "original" timeline, allowing him to go to the bench, where we see the aged Cap.
I could have made some huge mistakes, let me know.
ALSO: How did Cap turn the Space Stone back into the Tesserract, put the Soul Stone back into Loki's scepter, and (I still chuckle) put the Reality Stone back into Jane Foster? Perhaps the Reality Stone is able to reconstruct those housings? If so, you'd expect the Avengers to have done all that before Rogers stepped onto the time machine, and not expect him to try and figure it all out after he's already on the road.
ALSO ALSO: I reckon Cap didn't exactly live a quiet life in the alternate 1940s-2023 timeline. It still contains a whole host of threats and potential threats (and also friends), some of which his foreknowledge can help with neutralizing, and some that are outside his control. That timeline also contains its own Thanos, for example.. although it also has a Thor and Odin. As for his allies from Earth, anything he does, even his mere presence, might have a minor ripple effect that prevents some other individuals from ever being conceived/born in that alternate timeline.. what if there's no Tony Stark, Nick Fury, Jane Foster, or Peter Quill? Simple actions may easily lead to them never existing, especially since they're from the same continent/country.
"..history has shown the best defense against heavy cavalry are pikemen, so aircraft should mount lances on their noses and fly in tight squares to fend off bombers". - RedImperator
"ha ha, raping puppies is FUN!" - Johonebesus
"It would just be Unicron with pew pew instead of nom nom". - Vendetta, explaining his justified disinterest in the idea of the movie Allspark affecting the Death Star