I think Vile you just pointed out your own stereotype and the danger of them.
As you pointed out before stereotypes are fine if people recognize them as stereotypes. Yet here your own stereotype is presented, women being more comfortable with children, and you then attempt to rationalize that stereotype. This is where stereotypes become dangerous because everyone rationalizes them. Human beings are smart and most have the self-image of being intelligent. People recognize that holding a belief about someone else without some sort of supporting argument, no matter how weak, is stupid. Therefore people rationalize their belief and hence their stereotype. Here you have done this with women, unfairly putting on them the burden of natural child care experts.
Yes and no. I didn't say that the stereotype was a good one, I was just explaining why I can understand the reluctance some people would have with handing very young children over to the care of grown men without credentials. Rationalizing isn't a bad thing if used to recognize a stereotype, and then use that to remind oneself that sometimes they're about as wrong as possible.
Now, as for "putting a burden on women", your average woman (yes, average) has evolutionary tools that make it far easier for her to learn how to care for very young children. That doesn't mean men can't do it, and that doesn't mean all women are good at it, but if you only look at differences between brains (such as women having a large area of the brain for empathy, which is very important in the earliest years of a child), then if you take your average man and your average woman, the average woman will learn these things quicker and more easily. Now let me stress again that the average man and average woman does not exist. But there is a reason why these things come easier for a lot of them. If I look at the number of women I've met and how handling a child is like flipping a switch for them, while for a lot of men, it takes time, there is a disproportion there.
As a new father, I can speak of this myself. I'm perfectly comfortable with changing diapers, feeding, carrying, holding, putting to sleep, etc; but it took time, whereas my wife just picked these things up like they were nothing.
Again, this difference might not be massive, but it's there. And on top of that, I'm not advocating that woman should get all responsibility with children, not in the slightest. It needs to come from both parents, equally, and that includes the dirty diapers. So I'm not saying that this "burden" should go to women by default. I'm only saying these things come easier for more women than they do men.
Evolution may have shaped men and women differently, but nothing says we have to fulfill those or any other roles, but you also can't just close your eyes and pretend these differences aren't there.
You also ignored the fact that I stated the stereotype can be an understandable apprehension (it's one's own children, I think we're entitled to some irrational over-protectiveness, no?) only in the case of very young children. Once they hit the age of two, I've never heard of any real reason to think any guy can't be trusted with taking care of them.