It isn’t explicitly that the government (whether it is state or Federal) is giving money to the schools, it’s the students that are funded
I don't care what mechanism is used, the fact is that we have an incredibly efficient public school system, but it's struggling under the weight of its current demand with its current funding level. If our public schools are not funded to the point that they don't know what to do with their existing funds, private schools should not be seeing one cent of that money. Not when we spend years after deciding to build a new school just trying to figure out where to put it because the government can't afford to buy a site. Not when students of supposedly-free public schools spend thousands of dollars per year on various fees, equipment, and other school necessities.
They pay their taxes, they are entitled to the funding
No, they are not. I don't have kids. Where's my cut of that money? Where's my cut of the roads budget because I choose to cycle or take the train? Where's my cut of welfare?
Tax money goes toward funding government services. That's what it's for. You don't get a refund on any other service you don't use*. This should be no different. (* Okay that's not quite true. They do the same shitty thing with Medicare, and I oppose it just as virulently there—probably more so, considering in many cases you end up seeing the same damn doctors anyway.)
If private schools think they don't have enough money to run privately, then they could start by trying to run a bit more efficiently. Cut executive bonuses. Build facilities that meet the needs of students, not the wants of their marketing departments. Heck, maybe don't have as much of a marketing department.
You'll forgive me if a temper tantrum thrown by an offshoot of the world's largest paedophile protection racket does not convince me that we should fund them.