I have a bias where the period setting is concerned.
First Avenger is my favorite Captain America movie and one of my three favorite Marvel films. I'm sure folks will disagree on that one. But WWII is where Cap fits most naturally, and beginning with him in the context of the values of his era really sharpened his conflicts and relationships when he was brought into the present-day movies, IMO. If a director has a real feeling for an era, nostalgic though it may be, like Johnston did, then they can make something really atmospheric and attention-getting, at least.
The 60s setting is speculative, of course, so in the spirt of speculation: maybe the first part of the movie takes place in the 1960s but the bulk of it is in the present. It's a four-very-odd-fish out of water story, right? Hanks spends most of
Big as an adult, after all. It goes without saying that Marvel movies jettison the continuity of the comics without much of a second thought. So the FF have their Cosmic Ray experience but never go public afterward in the few months they remain their own time. It's one of Reed's attempts to fix things that propels them into the 21st century.
Reed Richards in particular would find "the future" about as confusing and distressing as
Time After Time's H.G. Wells did. Reed already
had a flying car sixty years ago.
Carry the
Time After Time thing further: Richards knows what a monster Von Doom is from stuff that happens between them back in the day. The FF arrive in 2025 to find him a solid international citizen and benefactor of the masses in his own country. He's a genius; maybe he builds, I dunno, EVs and social media platforms. And Reed's going to go nuts trying to warn the establishment that Viktor is a bad dude on the verge of some Bad Bad Awfulness.