Talking about programming-related courses, I can see the point of testing on a computer where one can run and debug actual code (that's how I had my programming courses) but I am not sure I get the advantage or writing code or pseudocode on a "basic word processor".
Moreover, for math or math-heavy courses (assuming most people with degrees here have STEM degrees, and many with at least some math) I cannot imagine how to comfortably write math in a word processor. Or use latex and not spend half the time troubleshooting latex, esp without internet access. So for some kind of courses at least, imo pen and paper for a timed in-person exam is the only way.
Otherwise def doable, but knowing how some universities function, I think the main problem would be getting the agreement and initiative to set such a computer room up. Getting some kind of consensus between professors that this is how (some) exams should be held and including it in the Holy Curriculum. Getting bureaucrats understand what it is about eg why you need these wired connections when the uni has a campus-wide wifi. Getting IT security agree with using old computers with lubuntu instead of their bloated enterprise windows "secure" OS. And if they are not connected to the internet how will they get security updates? How do we conform to whatever IT security rules are in place?
Writing on paper is much simpler, everybody can understand it and has been standard for decades at least. It can start tomorrow and be used in the interim while waiting approval for such a computer setup.