>There are hundreds of thousands if not millions of people who would definitely have more children if they won the lottery tomorrow.

If this were true (and I don't concede that it is), then you've just stated that the minimum bribe sufficient to persuade someone to become a parent is a lottery jackpot.

Mostly though, people who won lotteries would rather spend that on themselves. Demographic collapse is inevitable, and you'll die without even understanding why it happened.