The standard answer seems to be vouchers. Keep the free eduction, but let schools compete for students.
This is an often flouted myth. Football pays for itself and then some. There's more money in education, per capita, inflation adjusted, then there ever has been ever.
It's not a money problem, and it's not a sports problem.
It’s like saying home mortgages are free money because they’re also partially backed by the fed.
Your analogy to mortgages is right on, but you've drawn the wrong conclusions. Fed-backed mortgages absolutely functioned as free money. Because the risk of unqualified borrowers was assumed by the government and by opaque, byzantine, and corrupt securitization schemes, banks were incentivized to hand over money to people who weren't able to pay it back.
This was, in every sense that matters, "free money". And it led directly to a financial crisis that many millions of people have never fully recovered from.
The problem with this approach for public schools in the US is that there are multiple taxing authorities involved in determining how much $$$ gets attached to a student, and most of it comes from local property taxes.