That surprises me... why would they make newer versions that are less secure?
As for Mozilla (I wasn't even thinking about that because I'm partial to Opera :p), it would certainly be an improvement over IE or NN, but I don't think the chances of getting them to use that are very good for the reason that Choster stated. I doubt they'd want to try something in beta for all of the computer labs, even if it is stable.
I would have thought that a main section of their site not displaying correctly would have been reason enough. :p
Thanks for all of the other suggestions and things to think about. Anything else?