The reason the woman ends up at home more often is a function of who has babies, maternity leave, differences in age that flow to earnings in marriages, attitudes that are still quite embedded around role in society etc.
I'm sure you know that though.
Still, increasing numbers of couples are splitting it out, both going part time for eg.
IME there are men who stay on at the office to avoid Bath time, bed time etc. Ask them if they would like to be at home with children and do so housework etc while wife works, they would say no. No way.
Men know all this stuff is work. For quite a few, the set up of having someone at home to do all that stuff is what they want. They don't want to wipe arses, clean toilets, hang around at the school gates, source last minute costumes for plays no one mentioned.
Some people like all that some people don't. It is most definitely work though. Work that some men like to avoid by staying late at work.
If men wanted to be at home with their kids then it would be so, as they are the ones who set society up, and they are the ones who still, generally, make the rules.