That could be an interesting first use case: accepting partial or full tuition payment or maybe student fees or club/activity fees with this?

Yup, that could become big - Facebook started first in academias and first users were students.

Agreed, college students can be a good target user-base. Making payment easier/simpler/"more futuristic/cooler" could lead to buy-in/usage. Just recall how it was to interact with your university registrar/bursar...

Assuming they do, what use would they have for this coin afterwards? Can they convert these payments into salaries for staff? Equipment for laboratories? Repairs? etc. etc.

What would be gained from converting money into cryptocurrency to buy something you could just buy with money?