Hi Rod,
In the early 2000s, San Diego State University used random pairings for our Greek life events such as Greek Week. Since I was working in the Academic Advising Office and not in the Greek Life Office, I can't remember if Homecoming was random pairing or GLO choice. However, Greek Week was definitely random. SDSU has a large variety of GLOs and councils beyond IFC, NPHC and NPC. We also have the United Fraternity and Sorority Council (USFC) which houses the culturally-based, multicultural, and special interest GLOs.
The staff at SDSU's Greek Life Office was a big proponent of the idea that we are all Greeks *together* and supported measures which would bring unity across all of the councils. One of those measures was the random paring of GLOs from all four councils as teams for the huge Greek Week. For their part, the students were willing participants. I recall that the members would always seem to be having a great time during Greek Week - even when paired with groups they didn't really know. Yes, there is some talk of tiers at SDSU; but, not at all like other universities. I think we have a general culture of tolerance here in Southern California or maybe just San Diego. That said, it's definitely do-able.
Here is an older article which does a decent job of spelling out the Greek Week events:
https://thedailyaztec.com/21653/dail...k-for-charity/