Bill it is the government's and by proxy the rest of society's business. The very fact that in every developed nation it is the law that a child needs to get a basic education means we have already given tacit approval to the government being somewhat involved in the raising of children.
It would be both negligent and immoral for the government (and by proxy every one of us) to legislate that children need an education and then not be involved in what that education consists of.
I don't want to get all Dawkins here but there may be an argument for child abuse with that type of indoctrination.
Take for example (okay this is extreme) the case of Omar Khadr, indoctrinated in fundamentalist Islam by his parents and who ended up killing a US soldier in Afghanistan.
That this whole tragedy was the result of child abuse looks to be a valid argument with me.
It is a great argument. I disagree with it, but it's a great argument.
The problem is that it involves a balance between the needs of society and the rights of citizens, including the right to raise their children as they see fit; to include bizarre religious beliefs if they so wish.
In the USA, the government is prohibited from establishing a state religion, or from favoring one religion over another. It is also forbidden from infringing on anyone's right to worship as they please.
While I agree with you that teaching of religion does not belong in public schools, I agree for other reasons; I don't like your reasons.
I am not concerned with 'what' in this case; I'm concerned with 'why'. We agree what - religion should not be taught in public schools. We disagree on why.
If the 'why' it should not be taught is that such teaching creates a potential danger to society (your current example) or such beliefs are just wrong (your first argument) or such beliefs will damage the potential future ability of the child to become, for example, a biologist (your previous example), one sets the state up as arbiter of what religion is OK to teach and what religion is NOT OK to teach.
This is dangerous for several reasons.
First, the arbiter of religion possesses enormous power to oppress any religion at the expense of any other. It basically does set up a 'state religion' to the extent that the religion is 'state approved'.
Second, it directly leads to oppression of religion by simple extension of logic. If it is dangerous to teach a child radical Islam in public schools, it is just as dangerous to teach it in a private school, in the home, or in the mosque. And it is - I doubt many would disagree with your argument.
BUT - if the state has the obligation to restrict the teaching of such dangerous religious beliefs in the school, then they MUST also have the right to restrict it in the home and the mosque.
Freedom is not always conducive to an orderly society. Rights rub up against each other and cause problems.
I come down on the side of individual rights in such cases. Society has a say, but where it infringes on individual liberties, society must step back.
That means future Omar Khadr's will exist. Society has the right to protect itself against such people. That does not extend to denying Omar's parents from teaching him whatever they feel they should.
And frankly, I'm not even sure you can draw such a bright shining line anyway. Adam Gadahn's parents did not teach him radical Islam. David Koresh's parents did not teach him whatever it was he ended up believing. Same for Jim Jones, etc, etc.