Stargate SG-1, Season 4, Episode 6 has a variant of the loop trope, but everyone (including most of the protagonists, and everyone else on earth) don’t remember what happens, while two protagonists remember every loop until they are able to stop the looping.
They debrief the others who don’t remember at the end (except for the things they did when they took a loop off anyway!) - but they didn’t miss too much since everyone else on earth missed it.
Another fictional work - a book, not a movie / TV show / anime - is Stephen Fry’s 1996 novel Making History. The time travel aspect is questionable - he sends things back in time to stop Hitler being born, but no people travel through time. However, he remembers the past before his change, and has to deal with the consequences of having the wrong memories relative to everyone else.
He is not going to run for president again ever in a free and fair election in accordance with the US constitution; that would require changing the constitution in ways that the Republicans don’t have the numbers for, or at least interpreting the existing constitution in a way that is so contorted I don’t think even the most conservative supreme court judges could support it.
So in other words, he does not need anything from the American public anymore. He has no reason to care if part of his base opens their eyes to what he really is (at least, as long as at least 1/12th of the public will vote not to convict on any jury - but he can also self-pardon for anything except impeachment).
I therefore don’t think the kompromat theory holds much water today.
More likely, the Russians calculate that this is an opportunity to sow division in the US - they’d hope for a civil war as the best case. Supporting Trump, as a divisive president, was a start, but they wouldn’t want too many people happy with Trump either, so they want to make the haters hate him even more than is rational, and the sycophants continue to love him more.
Of course, the risk for them is that they make Trump want to support Ukraine to a greater extent than the US currently is, instead of the opposite. They probably calculate he is incompetent and nothing much will change for them either way. Trump is certainly installing yes-men who will be loyal to him but likely not the most competent leaders; this is an effective way to disrupt a government, but it is likely that a declining narcissist who has structured things to remove all dissent will not be at all effective in achieving outcomes that require complex strategy and coordinated execution. So I think they probably consider this risk to be acceptable.