Your second claim doesn’t follow from the first. It may well be a good idea–in fact, it is a good idea–but there’s no logical connection between legally requiring all parents to educate their children and ensuring that the requirement doesn’t become an impossible burden.
What would happen if we simply allowed that impossible burden? The same thing that happens where, for instance, housing is extremely expensive: the birth rate goes down. Most people have at least some idea of whether they can afford a kid or not.
Of course, some people would have kids despite being destitute, and we’d want them educated, so we’d have to account for that. The cost of these services would be much less than those for the population as a whole. It is the equivalent of uninsured motorist insurance.
One might posit a different argument: people should have the right to have children, and the presence of a law that obligates them to pay a cost they might not otherwise pay is a heavy restriction on their freedom. This is a somewhat reasonable argument but it’s not like we don’t do the same thing in other areas.
So I stand by the claim. Forcing parents to pay the real costs of their children–including education–would result in a massive decrease in birth rate. Western nations need to do all they can do to keep the birth rate at least at replacement levels, so we spread the costs around.