They need to be supported. They need to be funded.
I wound up sending my two sons to a private school (from an excellent public school) and the very first thing I realized is that the public schools are significantly underfunded. Even the very good ones. Not that this matters very much, but the tuition we paid still was less than the cost of educating the kids, and they depended on serious fund raisers.
The public schools can't use fundraisers. They are totally dependent on what each state gives to educate the kids. Plus, of course, they must educate all comers. They can't deny someone for any reason at all, unlike the private schools.
I was often quite unpopular when in conversations with other parents of the private school, I said that I should not be exempt from any school taxes whatsoever. I understood how incredibly fortunate we were to be able to send our kids to the private school. But just as those who don't have kids in the public school, we all MUST contribute to the public schools, absolutely. As it is, even if private schools didn't exist, at any given time only a minority of us have kids in the public schools. But we all benefit, and so we must all contribute.