You are proposing to offset economic inequality by degrading the education of the privileged. Your goal should be improve the education of the disadvantaged instead. Raise all boats, don't drain the lake. It's obvious that the United States could spend much more on education and do a much better job with the schools they have. Pinning any significant portion of the "huge cost" of the current system on middle class homeschoolers seems like sophistry intended to avoid the abuses of the hyper-capitalist, atomized system and ignores the actual problem.
> Every day hundreds of thousands of parents "break down" and government takes over for them. Government breaking down is a much rarer event.
This doesn't address the issue at hand. If a parent breaks down and the government steps in, that isn't incompatible with permitting homeschooling in non-abusive contexts.
> If you don't like the distinction between secular and religious, you can take any other strong belief. What if my parents are flat earthers? What if they are holocaust deniers? What if they hold some very strong political opinion? Or even what if they are some staunch atheists? They will just create a bubble around their kids.
If your parents are extremists of any sort, you're not going to mitigate their toxic influence or prevent them from harming a child by eliminating homeschooling. Most bad parents still send their kids to public school. Public school is not a remedy for abuse.