[sp332]

However, we live in an at least nominally secular society. That means that we don't show preference for or prejudice against religions.

I think this is where you're wrong. Freedom of religion doesn't mean secular, it means you get to have whatever religion you like. Consider:

The US government isn't a Catholic organization, so we won't give tax breaks to Catholic charities.

The US government isn't a Libertarian organization, so we won't give tax breaks to Libertarian charities.

should not lead to an expectation amongst the religious that they should receive similar treatment from private enterprise.

You really can't separate charity from the ethical principles that drive people to do it. Religion is a private enterprise.