In essence, it comes down to government use of power. Private companies may discriminate based on beliefs, because the positions they offer are not created by use of force, whereas all government positions are.

Put in a concrete, extreme example, consider if all teaching positions across the US required a statement of belief (say, in god). This would clearly be a violation of the establishment clause (really, tied to freedom of speech, but you could make another example that is more closely linked to the current situation). However, if some private catholic school required its teachers to believe in god, that would not be a violation, because the constitution binds the government, not private individuals or groups.

So, in a clear answer to your question, "why should it in this case?", it should because the constitution is a set of rules for the government to act upon, not for private companies.