Perhaps you're coming from a different cultural context than, but in my circles such universities absolutely are mocked and looked down upon. Some people don't even consider applicants from BYU, and other heavily religious universities because they don't want to reward such institutions.
> I know it's an unpopular opinion, but I'd argue that colleges and universities should have the right to police speech and expression: they should just be upfront about it. "Our house: our rules" as they say.
For public universities, the First Amendment legally obligates them otherwise. For private universities they already have the right to police speech and expression. It is, as you say, their house and their rules. They don't police (or rather they are very liberal in their policing) because freedom of speech and expression are central to an effective academy. Once universities start policing heavy-handedly, or on ideological grounds people start to doubt whether the ideas voiced are genuine or whether people are censoring themselves out of fear of retaliation from the institution. This cloud of doubt hangs overall the research published by that university, and the reputation of that university suffers considerably.
