Religious beliefs are, in some situations, given protection from government criticism because of the separation of church and state. For example, in a secular system government-provided schools might not be permitted to raise matters of religions. In a science lesson a teacher can use the value of the speed of light to indicate that stars are very far away to show that belief in alien abductions is almost certainly mistaken, but they are going to get into difficulties if they use the value of the speed of light to show that creationism is false.

I'm not saying that a secular government can't end up following a NOMA-style policy. However, NOMA isn't a guaranteed logical consequence of having a government that refuses to deal with religion. It could lead to a system of government that cuts out any and all religious influences in classes outside of RE ones.

Take that example above. Secularism means that creationism doesn't enjoy a privileged position in the classroom because there's no state support for it - they're separate. If the teacher refutes creationism, she's making a scientific claim, and is therefore doing her job. You provided an alternative way to deal with that sort of quibble yourself. You point out that:

...religious claims are also scientific claims and philosophical claims, even matters of morality. So, in principle, all aspects of religious belief clash with attempts to teach such subjects and legislate on such matters.

Deconstruct religion to the point where you've no longer got religion anymore - religion is a composite thing in any case - and secularism could just as easily be a tool for wiping religion out of existence. The separation of religion and state, in any case, simply means that no one religion gets to impose its dogma on people unchallenged. It does have a missing link, which is some means of distinguishing nonsensical justifications from sound ones (like in the whole euthanasia and abortion law debate), but this is more because it's simply open-ended what a secular government does to tackle this.

I think you're confusing "government has no religious influence in it" with "government has no influence on religion". The definitions of secularism simply say that if a citizen wants to follow Religion X, then be it on his or her own head. Their actions still come under state law.

Secularism is also highly undemocratic - if certain beliefs are 'unionised' - sufficiently organised to form large groups then those beliefs get protection from the state in a way that the beliefs of individuals don't.

I thought the idea of a democracy was that the majority got the clout. If a belief system has enough believers to be a force to be reckoned with, I'd say that's pretty much democracy at work.