So by excluding straight white students it helps build a more "cohesive" experience.

Brilliant.

... Yep... Just like a 5 year old.

Don't dismiss it.

If even one of their groups were for white-privileged brats only (or even those who self-identified as such), would ANYONE consider it a tool to help build cohesive unity within the campus?

I could care less about exclusive clubs.
But I despise the lies and those who agree with the lies in an attempt to justify it.

Tell me that enough clubs on campus are already too vanilla in their membership. By means of interest or exclusion, it really doesn't matter.
This club is an opportunity to speak about "their own" issues without some white dude (be they KKK or Bleeding Heart) listening in on their jive talk.
Fine. Run with it.
But don't expect me to believe their lying justification.