Your argument is interesting, but the law is well settled.
It all boils down to taxation -- public schools are funded by taxes, and we all pay them. That includes Catholics, Protestants, Jews, Hindus, etc., etc., etc.
There is nothing so galling as being forced to fund somebody else's religion. Jefferson and Madison recognized that, hence the Virginia Statute for Religious Freedom, and the First Amendment. In their day it was the Baptists being taxed to fund the Episcopalians.
You're free to believe whatever you want, but you can't use my tax money to proselytize your religion.
I fail to see how the notion that organized matter behaving according to laws as explained by the possibility of intelligent design amounts to "proselytizing" or "indoctrinatiiong" people. You must believe people to be wholy weak-minded. You don't trust them to think for themselves. You would argue against free inquiry, and that is hardly the part of science.
I also fail to see how "proof of the existence of God" impinges upon the discussion. What kind of "proof" are you talking about? Regardless, public schools are, like you said, open to people of all faiths and lack thereof. They can either be open to all points of view or they can operate unconstitutionally as agents for the state indoctrination of non-theistic principles.
The debate between theistic and non-theistic principles in public schools is not an either/or debate, but both/and debate. Otherwise shut them down.