I don't understand why, any more than I get the desire to keep putting Spider-Man back in high school in modern adaptations. It's not like the FF is a series that existed only in the sixties; it's been in constant publication ever since and has updated with the times. The sixties are only a small part of the series's overall history. So I see no reason to associate them specifically with their beginnings. Especially since both feature-film attempts to focus on their origins have been weak, since their origin is the least interesting thing about them.
To me, what defines the FF is that they're on the cutting edge of progress and discovery. They are to the comics what Tony Stark was to the MCU. I don't see how they work as a period piece. If they lived in the '60s, I don't see how you fit their adventures of great invention and discovery into the history of an MCU that didn't gain public knowledge of alien life until 2012. It would all have to be secret, and that doesn't fit the spirit of the FF, since part of what's defined them from the beginning is that they're highly public figures.
And if they come forward from the '60s into the 2020s, it robs them of their defining trait of being on the cutting edge. Even the most brilliant person on Earth, which Reed Richards arguably is, would be pretty much useless as a scientist/inventor if his knowledge were 60 years out of date. Okay, maybe they're from an alternate Earth in the multiverse where technology was more advanced in the '60s, but there'd still have to be a learning curve if they came into Earth-616.
So I can't imagine a more inappropriate and self-defeating way of integrating the FF into the MCU than putting them in the sixties. If it were a standalone continuity, maybe it could work, though I don't see the appeal. But putting them into the MCU proper that way would require stripping away many of their most defining traits.