Also, keep in mind that I don't actually need to prove those premises to show that my conclusion follows from them or is implied by them; an argument can be valid without having true premises.

RIght, but my point was that your conclusion doesn't follow from your premises (though it is evidenced by them). The reason is that 5 does not follow from 1, 2, and 3, and so the argument is invalid. It could be, for instance, that philosophy programs focus on producing learned rather than good philosophers because they are incapable of producing good philosophers over and above learned ones (suppose we grant my premise that there is no such thing as a good philosopher, for example).

I'm not actually contesting the truth of your premises or your conclusion.