I can see arguments on both sides.
I personally don't see anything inherent in the F4 that it must be set in modern day.
The strongest argument against setting it in the past, is that you want all of the F4 library of characters to interact with the modern day MCU characters.
Yeah, you could explain that F4 team was trapped in some type of time warp or whatever , but would Doctor Doom, Alicia Masters, The Mole Man, Galactus etc, also be trapped as well?
If they weren't , then several of the human characters would be in their 80s at the youngest today.
I would think that they'd want Fury , SWORD, Spiderman, Hulk etc to interact with an emerging threat of Doom, Galactus, etc as opposed to those characters being quite old today.
You want these characters and their supporting casts to be able to interact with the current roster of characters in the MCU as opposed to being isolated to the 60s.
To me , that's the real set back to making F4 a period piece, even though i'd be totally down for that.
The other arguments against a past setting really don't address the most obvious setback to having them in the 60s .