I think it's probably a bit of a mix.
However - sport is a 'men's issue'. Men talk about it, read about it, care about it. Half the fecking news is taken up with sportsball, and its seen as an acceptable topic for discussion.
I wonder if the whole premise of sport (sublimated aggression, competition as a playful means of fighting) means it's also seen as more acceptable for 'robust' views. Men can whang on for hours about what is and isn't fair, and they are far more relaxed about hurting people's feelings.
Women's dv shelters ... I'm sorry, this will sound harsh, but most people and especially men, just don't care. Most people think abuse is a very niche issue, that women who experience it are generally partly at fault themselves for getting into the situation. It's not that they don't care that men are accessing the service, it's that they don't care about the women using it in the first place.
They are also seen as a 'women's issue'. Women are, on the whole, firstly more 'be kind' and considerate of vulnerable men, and secondly more likely to attract attacks and opprobrium if they speak up.
Remember the woman who stood up and called 'shame' on Nicola Sturgeon?
https://news.sky.com/story/shame-on-you-nicola-sturgeon-heckled-over-scottish-government-gender-reforms-12758613
Exceptional, and exceptionally brave. Look at what happened to Roz Adams.
https://www.bbc.co.uk/news/articles/c6248z383x7o
As has been said before, women on this board are often especially gender non conforming, and a big part of that is that many are less concerned about being 'agreeable' and displaying the 'masculine' traits that distinguish us from a doormat.
Another issue is visibility. Sports is full of photos, we have astonishing visuals of Lia Thomas, that weightlifter bloke, Fallon Fox, the lanky Aussie footballer, the Italian/Irish bald footballer, Iman Khelif. These are (aside from the fat weightlifter) big, burly, healthy, muscly, men wearing lycra and sweating and being photographed and filmed from every angle in bright daylight.
For obvious reasons, women in shelters are not in the papers, not talking about the issues. Nobody is photographing the men accessing women's shelters unless they are subsequently on trial (Katie Dolatowski, Barbie Kardashian, Cameron Downing, etc).