You choose your religion.
I "chose" my religion as much as I "chose" my native language.
My religious institution was very good at binding me psychologically long after I stopped to believe.
Anyway, not to allow religious colleges latitudes regarding faith is an encroachment on freedom, and the bashing of such institutions shows intolerance--
A normal employer is not allowed to discriminate against atheists or believers including Belmont graduates, so why are religious colleges entitled to more freedom than everyone else? If they want to employ a priest, that is one thing, but if they employ a teacher for basket weaving and a cleaner, that is another thing entirely.
I think it is quite fine to show intolerance in the face of intolerance.
For the record, I would certainly speak up against the idea that religious belief makes one unfit as a teacher or researcher, but in the places I have been this is an outlandish scenario. I promise to remember if things are very different in thirty years time or a different place, and I will easily point to the religious people who have disagreed with me on most political and religious issues, but have supported me as my teachers regardless.
The exemption on religion in AA/EOE rules for religiously-affiliated institutions is the same one that applies to churches (I'm going to presume that, unless like amnirov you viscerally object to churches, you're ok with, say, a Catholic church not being required to hire a Muslim to be the priest). It's based on the idea that the religious identity of a place is essential to its mission and existence and is therefore entitled to an exception. A so-called "normal employer" has no fundamental reason to impose a religious test, and is therefore not allowed to do so. A church or other religious institution does have that fundamental reason.
Yes, I know, the same arguments have been made about women at Augusta National (there, the argument is utter nonsense, but that's a different thread), and in the past had been made about nonwhite people in various places.
I'm not religious, but I'm wholly untroubled by the exemption from the religion part of AA/EOE rules for schools such as Belmont. They're not lying to anyone about what they are. They're making what I presume is a sincere effort to adhere to the parameters of their missions. Their religious identity is part of their missions.
By the way, I expect that if you founded "Atheist University," and made "The Rule of Reason" its mission, you'd be perfectly free to exclude religious believers under exactly the same rubric with which they'd exclude you. But even though the US is pretty religious (I'd certainly agree that it's excessively so), the fact is that there are hugely more secular institutions than strictly religious ones, and there are indeed places (many, many of them) where the overtly religious are unofficially and illegally and covertly discriminated against. At least Belmont University is telling the truth about how it discriminates.