Or, your children will eat your other children, or possibly you, and maybe both. Regardless they are very unlikely eat my children because yours are far more likely to end up sold into child slavery or prostitution by virtue of your locale. Which location is also so extremely removed from my children that any interactions at all are unlikely.
I'm not opposed to having large families, I come from one myself. And I don't consider the idea of restricting procreation rights lightly. But in the hypothetical scenario of eliminating aging it would likely be a necessity at some point, even if that point is centuries away.
By collecting genetic samples from each person they can then later have children if the opportunity arises. Sterilizing someone is hardly the same as nuetering. Sterilizing merely removes the ability to procreate via natural means. Nuetering would actually remove the organs, like testes, which fill other purposes also which would be unnecessary.
People that don't go with the anti aging treatments could procede to have their replacement kid, but if they want more they'd need to purchase lottery slots from others, or move elsewhere, like you said. In those other places were such laws aren't the standard you'll probably see an eventual poverty state with overcrowding and lack of resources. At that point depending on the situation wars will probably ensue over resources between the nations that didn't restrict breeding, and those that did. The winner could be the side with the most people but it's unlikely as they'll likely be very resource poor at that point.