Eh, I don't think an 80/20 thing like this has anything to do with what Democrats do or don't do. It's a cultural thing, and it's going to take a while before that needle moves.
There may be a lot of things Democrats are doing badly, could do better, are dropping the ball on, etc., but it's not like they can change the culture overnight. It wasn't really the Democrats that moved the needle on things like gay marriage, either. Probably very similar breakdown on that as there is on sports. You'd hear stuff like "well, we shouldn't arrest them for what two adults do, but why do they want to call it marriage? Why not create a separate but equal institution and call it 'civil union'? The term "marriage" belongs to xtians, etc. (LOL, are they not merciful)". But eventually the culture generally shifted.
You hear things like that now on trans - people that understand it's a human rights thing. But are not all the way there on sports. And specifically on that. It's not exactly the same, but it does rhyme.
That's why I don't think it's worth Democratic politicians' time to play the Republican gotcha game on this. They should acknowledge their values - human rights are to be upheld, why are the freaks in the Republican Party so obsessed with the genitals of children anyway? How do sports even make the top 50 of the list of concerns for the American people? Mock the conservatives for their trivial and hateful obsessions, and move right on to issues (affordability, Iran war, assaults on human rights across the board, AI and AI data centers) that they have a clear lead on. Talking about "men in women's sports" is just accepting the Republican framing on it as well.