Also, since many private colleges and universities are Christian, and receive funding, NOT giving the same for schools founded by and for people of minority religions, you would implicitly be saying that the US is a Christian country. It’s not. The establishment clause just means that you don’t discriminate between religions.
Just a quick point of clarification, “recognition” isn’t an abstract concept, but would allow the club a right to funding and use of university facilities and resources. Events held by the club would be considered university events. All of which would, in the perception of Orthodox university leaders, be violations of halachic law. I’m not endorsing the behavior, just explaining that the impact is a bit bigger than just "oh we don’t want say there are GLBT people here.
As long as they get federal funding, they have to abide by non-discrimination laws. Even if only a small percentage of their student body is secular. The same is true for any religious institutions that receives federal funds.
Being religious is no excuse for violating the law.
Well those private religious universities still need to follow federal requirements, or else give up federal funding. [Edit: As you just said in follow-up] I went to one long ago. (For physics & comp sci, not for the religion - and because it was academically good, and I got great scholarship money).
In that day it was about things like Title Nine requirements on gender discrimination and educational/athletic funding. The school was affiliated with a pretty milquetoast, liberal denomination, and it didn’t resist TOO hard. It’s just that the big alumni donors weren’t happy when their athletic donations were forced to be shared equitably with things like women’s basketball and tennis. A shocking concept at the time! What does college athletics even MEAN in Texas except (men’s - do we even need to say that?) football??!!! LOL.