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It's not a radical idea, or really even unpopular, to incentivize child birth through tax policy.
Insane initial implication that having children is a duty we all innately have to society aside: you can't tax an age bracket differently let alone at point of sale.
If people take contraceptives, it's because they don't want to have kids at that moment. Forcing these people to have kids will create broken parents and broken kids, who go on to become broken parents themselves and perpetuate the cycle. Most of modern society's problems can be traced to parents who should not have been parents in the first place. We don't need more of that.
Industrialized free market economies are incompatible with high birth rates.
>How did even less generous free market nations have high birth rates in the past?
Because those nations were not yet fully industrialized nor did they respect basic human rights for every individual. Even in the U.S. it was not possible to have a no-fault divorce until the 1970s.
>Don't you think being forced to choose between having a free market and having high birth rates an even more extreme idea than the one in my OP?
Do you also think stock buybacks is an "extreme idea"? It's the nature of every business in a free market economy. This is why they punish employees who take leave instead of continuing to work.