As someone who works in Student Administration with an expensive and broken system, I often wonder why we can't just get the IT students to build us a system as part of their Final Year Project.

That's just cause they have better things to build.
If you really want them to get it done, gamify the process

In my university, it was done.

It has a problem, the student leaves the university and everything is legacy code without documentation. The student of the next year normally creates a new API to the db because he says that the old one its horrible. And the cycle continue.

(And ofc the application have multiple security issues, can't run more than 24h straight (needs to reset))

That's how my university ended up with a broken and expensive to maintain system..

Build one year.
Refactor another 5 years, then maybe.