Most in the suburbs don’t live in a “nice home”. There are more expensive homes dictated by tiers where you have more square footage and maybe a pool or something but these homes aren’t particularly nice either and aren’t well designed in the interior or exterior. Typically this is because there is too much space to do anything neat or useful.
Of course you’re right that many states fund schools through property tax and why I’m bemused by Republican efforts to initiate school voucher programs because based on typical policy relying on home ownership and property taxes to fund schools “keeps the undesirables out”. Once you go the voucher route you can’t do that anymore. Of course you can say well they’ll charge extra fees and such, but that’s assuming that families in the suburbs can afford extra monthly payments on top of everything else and I just don’t see it.
On the other hand vouchers can be a nice tool. Why should someone be forced to go to a bad school because they can only afford rent or a mortgage in the cheaper part of town? Maybe every kid gets a voucher to attend any public school they wish and the schools have to adjust. Finding a way to create competition within the public school system should be examined. Districting and property taxes for schools are probably regressive, especially for a child who has very little agency in the matter but has the most need.
The U.S in it's infinite wisdom is great at siphoning money away from people who have the least of it.