There is a difference in fertility age, but men don't start leaving STEM academia in droves when they approach their fertility limits- only women do. So the part about it being urgent to have children at a certain age may be a biological difference, as you say, but the part about leaving academia to have children is not a biological difference.
Or, to put it more simply: why don't men leave their careers to have children at any age?
Pregnant women can work except in the last stages of pregnancy and the recovery period after birth takes at most a few months (normally a month and a half should be enough).
But even assuming the whole business of giving birth took a whole year (the full nine months for carrying a baby to term and another three to recover) there is no reason that this has to be the end of a woman's career for good. It's one year. People take one year sabbaticals and go and gawp at Machu Picchu. It's not the end of the world, especially if you're young and even in a fast-moving career like STEM academia. it's not like you can't ever read the latest publications again because you have to lie in bed for a few days after giving birth.
The care you say men must give to their wives lasts at best a few months of a year. After that a man is free to leave work to raise the kids and the woman can go back to work, if they so wish.
In an easy pregnancy the woman may work until the end. In Germany for example women are expected to work until 6 weeks before the birth.
Then there's 8 weeks of recuperation after birth, paid by the health insurance which are more or less mandatory and it would be stupid not to take them. Even if the body's not fully recovered at this point, the woman can work.
The question is, why would a mother leave her newborn in order to work, when the outcome will be worse both for the mother (weaker bonding, working through recuperation) and the baby (missing its mother during the day)? That's why it's common here that women either take one or two full years of parental leave or they stay at home N years. This can also be and is mixed with part-time work.
In any case, that would be an easy pregnancy. Childbirth over 30 becomes harder and over 35 it's classified as a high-risk pregnancy.
"After that a man is free to leave work to raise the kids and the woman can go back to work, if they so wish.
So why don't they do it?"
One reason could be that by the time those months (in reality a year) are over the mother's better at caring for the child than the father.
I've also seen that women tend to like spending time with their babies(!) so they prefer working part-time or staying at home.
Finally, it's often the case that the husband earns more. Women also tend to prefer men which are successful.
In the case where she earns more, it makes economical sense for the husband to work part-time or stay at home. I read an article about such a case a while ago, and being a stay at home dad looked like a sad existence: besides the social disdain and emasculation, that person wasn't even welcome to spend time with other parents (i.e. mothers), because they were looked on with suspicion.
Or, to put it more simply: why don't men leave their careers to have children at any age?