I'm sure someone will say students build LLMs all the time, and this and that, but there's no reason these universities cannot build their own LLM "labs" and work on a common LLM that their university builds, maintains and students and faculty can manage, and release, maybe even sell API access to.
It would be a key selling point for any student who wants a serious job in AI to work on such a project too. Universities could easily sell access to pre-hosted models mostly at-cost maybe a smidgen on top of costs, and fund the project as it gets better over time.
Any university doing this would prepare students for real software jobs mind you. Maintaining, building and supporting real infrastructure during your college years sells better than no experience.