To expand on the second point a bit more: one of the virtues of the department model is that (most) departments teach a particular way of approaching problems,

To expand on the second point a bit more: one of the virtues of the department model is that (most) departments teach a particular way of approaching problems, a "discipline" if you will. They teach you to "think like a [computer scientist|chemist|psychologist|etc]". Certainly, there's often a great deal of value in breaking down these disciplinary barriers and attacking the problem holistically - which is why many institutions have interdisciplinary programs and the like. But there's also a value in fields of study retaining some identity as disciplines, with particular methodologies, mores and so forth.