Depends on how it's handled. While I think your version is the most likely outcome, especially considering the creative team are a bunch of aging Gen Xrs, the reason the original escaped the pitfalls we are talking about here is that Austin's 60s sensibilities are the butt of the joke, not the advancement of societal norms so often decoratively labeled as "political correctness". The movie is about the character learning to adapt to the times, and not the character demanding the times return to the 60s status quo.

Really though, I think they already sort of made the movie we are talking about here, and it turned out fantastic. The 21 Jump Street movies were basically what we are describing, just without time travel shenanigans. Channing Tatum's character, who was hot shit at his high school in the 90s, returns to high school 20 years later and finds that the things that made him popular are no longer cool and he has to learn to adapt to changed circumstances. Obviously that one is specifically satirizing the change in school culture post-90s, but it works on a general level too I think.

Maybe if he has a new partner who's your standard straight-laced FBI guy, and Austin sees him kissing a man and thinks it might compromise the mission only to find out that's his partner's husband. Maybe have Austin meet the guy's kids first, too. I think it could be done, since not everybody from the 90s had that hard a time with it.