Kids in a denser neighborhood with easy access to public transit have much better quality of life on average. Parents who move to the suburbs for the benefit of their children might just be projecting their own desires.
As for “aeons of human nature”: for most of prehistory and then most of history, most people were either living in isolated tribes in the forest, or as rural peasant farmers. Not sure how either of those is relevant to considering parking requirements, building height limits, etc.
There’s no inherent reason for class sizes to be different between urban and suburban schools. Often suburban schools have more resources because the USA spends more money on public services in cities with wealthier residents. That’s unjust public policy, but nothing directly related to population density.
Hell, my dad was only a telephone lineman for Ma' Bell, and he could afford 5 acres of land and a house in (our city's version of) the burbs.
Regardless of future changes to zoning laws, not everybody wants to live in big cities. There's a strong appeal to live in a house of your own, where your children can play in the grass with their house pet. Where they can go to school with a class size where they have a chance to actually know everybody in their class.
If you really want to increase the density of cities even more, people will either require a) no choice but to move to the big cities, or b) a fundamental shift in thinking about owning their homes and raising their families.
And please remember that you will be working against eons of human nature when attempting to create that fundamental shift.