This is far from universal. In my state (California) the cost is distributed among all employers as part of an “unemployment insurance” scheme. Everyone who employs someone pays a small amount each month to cover things like family leave. (Yes, labor economics tells us that this really means that all employees are paying this amount.) but it’s not all taxpayers, and it’s not a progressive tax (meaning everyone pays about the same, rich people don’t pay more as they do for other government programs).
Given your other comment, though, it sounds like your concern isn’t “taxpayers” paying for family leave, it’s you. So yes, ultimately, you, productive member of society, might be paying for other, “unproductive” members of society (as defined by you) to have kids.
You sound grumpy about this arrangement. Of course, as an upstanding citizen, you’ve never taken advantage of incentives or benefits paid for by other taxpayers, right? And of course you’ve never had the assistance of a government program, or it was different because it was for something “not in your control” — unlike having kids.
This attitude rubs me the wrong way, hence my less-than-civil response. It’s like the people who hate public transit because they only drive, not realizing that public transit takes other cars off the road and improves their own commutes. Talk about cutting off your nose to spite your face.