I'm very far from a red-pilled anti-feminist but I do think this is a hugely overlooked problem. Perhaps too much of an aside but I was lucky enough to be able to take an extended parental leave when my child was born, being a "stay at home dad" for the best part of a year. It was incredibly hard work, incredibly rewarding but also a real eye opener at how society still doesn't know what to do with a full-time parent…
The fact that you need to apologize and explain why you care about issues that might affect men is telling in and of itself. The American Zeitgeist around gender issues is distorted to dehumanize men and alienate anyone who even dares to recognize this fact. I'm a pretty hardcore liberal and I I'm totally astonished on a daily basis at the hate in vitriol directed at men that go without comment, let alone castigation…
I'm conflicted about that. Not denying that what you've said is true but I specifically chose the term "red pilled" because there are huge toxic (predominantly online) communities that are openly misogynistic. e.g. in theory I'd like and participate in the "men's rights" subreddit, in reality it's a pit of toxicity. Some of the frustration there is justified (family law definitely seems biased against fathers) but the result is not positive.
I'd also add: a lot of this societal pressure against men being accepted as stay at home parents comes from men.