You need to look at the inverse of this though. If people want to be able to ensure that people without the financial means to participate cannot attend, all they'll need to do is attach a religious label to the school and that will ensure that the funds aren't able to be accepted. Creating an exclusivity by proxy.

The other issue is the assumption that public funds aren't supporting any religion already. This is a loop hole by claiming that a religion doesn't have an actual deity and therefore is exempt. Meaning, you can teach anything you want as long as there wasn't a deity involved.

Mindfulness, for example, comes from Buddhism but is widely encouraged in public schools. It's only allowed because it doesn't claim a deity. The "Church of Satan" goes out of it's way to claim they don't believe in any god. Hedonism goes back to the days of ancient fertility cults that were typically associated with some type of idol worship, but as long as you don't publicize the idol it passes the "no deity" test.

The "veil of ignorance" definitely covers use cases along with this. There are a lot of parents who are very aware that belief systems are being taught in schools, deity or no.