My daughter didn't pay anything to have her babies, either, because she was low income and qualified for medicaid at the time. And I get all the health care I need, because I'm of an age to qualify for Medicare.
In the United States, taxes aren't spread evenly across all income groups. Benefits aren't either. For a Finnish model to work, all groups would have to pay taxes and all groups would have to benefit from those taxes. That's a hard sell for both those who don't pay any taxes and those who don't get any benefits. Right now a large contingent of people pay no federal income taxes at all.