You're correct in that taxpayer money does supplement most universities budgets. However, taxpayer money mainly funds administrative costs, some operating costs, and faculty/staff salaries. At most universities, the majority of research funding comes from outside sources. The reason Federal grants allow PI's to patent their discoveries made with Federal dollars is that there is extra motivation for innovation, and eventually (in theory) the invention will become part of the public domain when the patent expires. Whereas a useful new pharmaceutical usually has a lifetime of decades and therefore yields benefits after patent expiration, advances in technology are typically useful for less time than the patent lifetime. Thus, I think the Feds intentions aren't quite fulfilled in those cases.

One point that no one has made thus far is that the plaque is more likely to be a beaver than a rat, as the mascot for both MIT and Caltech is, in fact, a beaver. If you look closely, you can see a beaver tale on the plaque.