My biggest quirk about the universities and institutes I've worked for so far is not the way they deal with grad students or postdocs, it's about seemingly 'unimportant' issues. Things like large, noisy, shared offices, lack of whiteboards, unnecessary bureaucracy, and lack of paid linguistic editing. At universities money tends to be saved at the wrong end, private businesses are much better at creating productive environments. Especially the constant lack of office space is hampering productivity. The priorities tend to be wrong (outside of a few centers of excellence), focused on getting as many researchers as possible. Instead they should spend a bit less on human resources and a bit more on a good work environment.
At least that's my personal experience and opinion.