This comment is helpful and relevant:
https://news.ycombinator.com/item?id=22080563
While women experience a "parent penalty", the one men would/do experience is much more severe because rather than being seen as necessary or expected, it is a character judgement: you chose to walk away and didn't have to so you have demonstrated you are wiling to be a traitor. (fair or not, this is the judgment).
If one is to advocate for men "stepping up", one must at least understand what they are asking.
"What, just step in front of the cavalry charge, they'll go around you don't worry you're just chicken hehe checkmate".
And while it might be unpopular to suggest this is so, I agree with the parent poster that the first thing that seems to get lost in these discussions is that taking care of children is not a burden. Work, yes, hard, yes, a burden? Only if you didn't want them in the first place. I say this as a father of 2 who sometimes sacrifices work to take care of my kids. It's a privilege the days I do so and I am currently undertaking a Herculean effort to modify the shape of my career and my income in order to do so more often because I love them and want them brought up right instead of leaving them in the hands of some daycare. Burden? WTF?
And it seems to be an absurd argument: "Caring for the children is a burden that holds back women therefore men should do it so women can work." Wait what? How does that fix anything?
This is why I largely stay out of these discussions. It seems to be impossible for them ultimately to be realized as nuanced. This problem is hard and a balanced, equitable solution is multifaceted. To state the solution as "well men should just...." in any context whatsoever does a disservice to the discussion.