There is one other thing to consider - here you have a for the most part openly gay, die-hard arch Conservative Republican. His sexuality was an open secret in the GOP, and nobody much cared because he was "a man's man", and his politics were good. There's a reason why the man is a villain in many of Broadway's more riske' LGBT fare like Angels in America.
The man's sexuality is actually kind of a fascinating topic. For all the claims he was deeply closeted, again, he was very open about his taste for men, and just as vocal about his disdain for - and mind you, I'm quoting him - "fruits and pixies". I kind of wonder if he might, especially as AIDS impacts the LGBT community the way it did in OTL, might pioneer some kind of drive for "respectability politics" for the LGBT community, or try to peel off a chunk of the LGBT vote for the GOP. Mostly for that aim, I would think, he was a party man through and through.
I mean, you regularly have between 20%-10% percent voting for the GOP who is actively hostile to LGBT issues. I wonder what the impact of having a GOP more open to the more conservative elements of the LGBT community may have on politics or gay rights.
I can think of a few instances where a low-profile glbqt person was known to have some sort of affiliation with conservative political groups. In the one that comes to mind(from Canada), the woman in question was defended by a right-wing journalist(who had apparently also retained her services as a babysitter) with something like "The difference between her and all those activists you see trying to push their agenda on everyone is that she keeps her sexuality to herself, doesn't make a big deal of it etc." And even then, the journalist didn't actually name the woman, just wrote something like "There has been a lot of controversy about a particular individual who blah blah blah..."
More recently, also in Canada, there was a VERY high-ranking Conservative cabinet-minister, with a burly, macho image, who was widely known(even among the general public) to be gay, but never actually came out publically. And this was in a country where people are supposedly more laidback, don't care about someone's sexual orientation, and even the Conservatives themselves had allegedly made their peace with gay marriage etc.
Long and the short, it's one thing for everyone in Manhattan to know that a local GOP bigwig is gay, and be cool with that. But, as of 2019, I don't think the Republican party on a national level is going to embrace an openly gay man, no matter how much he otherwise claims to despise "fruits and pixies".
I also think it's important to keep in mind what a non-entity Cohn really was outside of NYC. I think the only people who gave him any thought would be political junkies, and people who took some interest in what was happening in New York.
(For the record, the gay cabinet minister mentioned in my earlier example is NOT Jason Kenney, whose professed chastity is ostensibly religious based, albeit subject to widespread innuendo. The guy I'm talking about is someone about whom people would just casually say "Oh yeah, he's gay", with nothing controversial about the assertion at all.)