It may be worth noting that, at a lot of schools CS was (is?) part of the math department as opposed to engineering. It's plausible to me that as CS became more about programming (and brought a lot of associated cultural baggage with it), people who were in CS essentially for pure math reasons switched out of CS.

I wonder whether there was a point at which computer science departments began to become independent of math departments, and when that occurred?

It seems to vary a lot by school for historical reasons though I haven't studied the subject. Some schools it's been connected to electrical engineering forever. Other schools it grew out of math. There are areas of EE that have very little to do with computers, especially decades ago, so I suspect that a lot of it is historical happenstance.