That is, I'd propose we let some government pay the maternity leave, and finance it from income tax.

Though, I would probably fund it via a payroll tax so it could remain employer-paid. I don't think it makes any difference to the economics whether you tax the employee or the employer, but changing who pays is an additional decision you'd need to explain and that would be attacked by political opponents. A payroll tax is a smaller change and equally effective, so it's the better option.

This would still exist in this world. You're still down an employee. So you don't hire women or married men.

Also, it doesn't actually happen.

How do you accomplish that without violating the right to privacy? If a man keeps coming into work and his partner is pregnant, nobody has to know.

The disadvantage women have is that they can't hide a pregnancy forever. Men can go to extreme lengths to hide their relationships. Although rare, it is not unheard of for a man to have two families at the same time, each one hidden from the other, and maintain this state of affairs for years before getting caught (otherwise we'd never hear about it).

The fact is, an employer has no right to know what, if any, relationships the employee has, let alone the pregnancy status of them.

The fact that employers discriminate against pregnant employees is the problem we should be addressing directly.

The tax forms might be a key: Require everyone to be honest about their underage children, whether or not they are dependents, and give the option to claim fewer dependents on their tax forms.

Yet another option is to handle it much like FMLA. The employer doesn't get the right to know exactly what is going on and the doctor can use medical codes instead of plain English. My ex's psychiatrist did this when he needed time off for a major mental illness (he later went on disability for said illness). The employer doesn't have to know, but he'll need similar notice.

I would hope a reasonable person would never advocate for such intrusion.

This is a draconian idea. Besides, employers don't want it! If they didn't want you coming to work, they would lay you off. Otherwise they're going to look the other way. Families living on the edge will do anything to avoid this problem, including leaving the father's name off the birth certificate, if it comes to that.

Perhaps increasing the scope of the worse hire will make discrimination impractical, but it is still going on while it's at roughly 50%.

You can somehow make companies unable to discriminate by relationship status, and that would push it into "age bracket" discrimation.

I believe the problem is more fundamental and a cultural solution is required.

It's all about the culture. Get the culture right and the decrimination goes away.

I think we should strive to have equal maternity/paternity leave, but let’s not deny the origins of the current reality.

When I express my sadness that I have to work and can’t be with children all day long (same complaint my father and grandfather had), my wife often jokes that I should find a husband. It’s funny, but it’s ultimately true. Men are simply better able to do the sorts of things I take on in my family and women at the things my wife does

Paternal investmemt has at most 6M years (time since split from chimps who have almost none)

(as it happens, fathers can breastfeed, albeit not literally. Mothers can pump breastmilk for fathers to feed to the child, or they can use formula. It's a perfectly viable way to raise a child, and people do)

My point is that OPs argument holds some water. Notably, it suffices to support the statement 'women are more likely to want to be care-taker than men'. There remains an argument about magnitude, and compensating effects, but there is a difference.