I agree it's a stereotype, but also true and exhausting and pissing off a lot of women. It happened to me, and like the OP I thought I would never be with a sexist misigynist arse, and pre-kids I didn't think he was but... he was.
Why do they let it happen? I think there are lots of reasons, to do with the patriarchy and to a much smaller extent biology.
When you first have a baby you are physically tied to that baby for a while, and it's understandable that that is your role as a mum as they just came out of your body, may be breastfeeding etc. Then it's very easy for that to extend to anything to do with the kids being your role to be in charge of, and that pressure being put onto you by your OH and by society in general.
Most women I know, including the feminists, if their OH refuses to take their child to a birthday party or sort out stuff for school or do school pick ups, will do it rather than neglect their kids to make a point. If their OH fucks off for an all-day bike ride or doesn't come home from the pub when he said he would and then has a hangover all next day, most women don't just walk out and abandon their kids so it's fair. And so the inequality is entrenched and he just does it more.
You can insist you can plead, you can nag, you can remind him endlessly that it should be fair and you do way more than him. And I did. But you can't make him take responsibility if he just won't.
You can leave - and I did. But guess what, now I'm a working single mum who has the kids most of the time (as I and they prefer, because he is not great with them) and sorts out everything they need, has less disposable income and is more exhausted, than my ex who has a higher salary (because he refused to go part time, so I did, so my career took a hit and his didn't) and loads of free time to do what he likes.
But what happens if a woman leaves her male partner with the kids and sees them once in a while? Massive moral disapproval, shock, how could she do that. But when men do it, it's normal, it's just what happens, they even get praised for bothering to pay basic maintenance and seeing their kids at all.
Can't you see that the inequality is really hard to avoid?