You may be right in general, but none of that is a good excuse for a transactional sex life.

If I wanted to exchange services (labor) for sex, I could simply take the money earned from labor to purchase it from a prostitute. That is not what a marriage should be like.

Wow. What an incredibly sexist comment.

women tend to

I'm gonna stop you right there dawg

You are making a lot of sweeping generalizations that are wildly inaccurate. Some of those statements (hell, maybe all of them) may be true for certain socio-political subgroups of our society, but I absolutely do not agree that that’s the dynamic through which most heterosexual people view their partners (or more accurately, the idea of a partner).

You’re basically just regurgitating the “atomic family” ethos from back in the 1950s.

Reminding him to care for his kids

by treating him like a kid will not help. Yes, making clear that you expect him to share the care work with you is important. Making rules together can be a way of doing it, but he needs to do it because he is the dad and her partner and a reasonable adult that takes their responsibilities serious, not because he wants a BJ at the end of the week. They both need couple therapy, because he isn't a responsible adult and she infantilizes him on top.

Unfortunately, women tend to want partners, men want caregivers.

Thank you. Nobody's seeing that. All the comments saying the woman is mean, instead of talking about how irresponsible the man must be that he needs a reward system to do what he should be doing on his own* for his family.

*I'm not sure if it's the right expression. I mean by his own volition and out of responsibility.

Edit: I won't acknowledge the rest of your comment because, honestly, it got confusing.

I'm really surprised this got downvoted... don't get me wrong, I downvoted as well, but... this is not what I expected.

Men tend to trend more irresponsible, women more neurotic, in my experience. There's plenty of exceptions, but on the whole that's what I've seen. Neither is good, both can collapse a relationship. In straight relationships this can result in women taking on everything. Even where she's overcome (or not originally had) any neuroticism, a sufficiently irresponsible man can still put the problem on her shoulders.
What I think you're omitting is that this can happen in reverse.

Even when a man overcomes (or didn't originally have) any irresponsibility, a sufficiently neurotic woman can still put all the problems on him. He has to pull the tasks away from her because she thinks only she can do it 'right'. Only then can he pull his weight. But he then must also do the dance of convincing his partner that he's doing a good job, or she'll just feel compelled to do the work again herself.

Of course most relationships are somewhere between these extremes. And some even see the roles reversed. People are, of course, extremely diverse. But this is a common pattern I've seen.