Philosophy is why we have democracy today with checks and balances....over millennia people discussed and argued about the best kinds of governments to have, how to find good leaders, and how to avoid problems.

That's just one example out of many.

Philosophy provides no basis for objectively deciding what is the best system of government or who would make the best leaders. If it did then why are there competing systems that are at least as effective at governing a population, even if not as popular with the population?

There's no objective definition of 'best'

I don't think philosophy is very useful, except for having a good understanding of logic, which we don't need a full liberal arts department for. That said, there's no denying Bill Nye obviously doesn't understand what "I think therefore I am" means. For one thing, it has nothing to do with "thinking about existence" or whatever he said.

I studied philosophy for a year, and I find it to be one of the most universally useful things I've studied. It's the process that is useful, not necessarily the content. The subject matter which you apply it to varies greatly in usefulness, but the process of thinking and deconstructing that you acquire as a skill while doing it is tremendously helpful everywhere else.

Yes - and I think you hit on why people who don't do Philosophy fail to understand it. They see discussions about Metaphysics and think that's the point, whereas the point is how you justify positions, what's a good argument, what's bad. All of Metaphysics is useless, for example, but knowing WHY it's useless is invaluable =)

The parts of Philosophy that everyone should know and study, especially in science and engineering fields, are often part of a Philosophy department, but they aren't the controversial parts. These courses could easily exist independent of a Philosophy department. My Philosophy of Science course, pertinent to my field of study, being offered by the same department as my Hellenistic Philosophy course, not pertinent to any field of study except Philosophy, is silly. Same goes for my Discrete Mathematics + Symbolic Logic course.