Should There Be Such A Thing As A Public Speaking Degree?

Yes.  That's a good idea.  They could use a Joe Biden speech on how not to speak in public.

I can certainly see the point of a short subsidiary, taking in the ancient Greek and Roman orators and discussing the psychology of winning over an awkward or hostile audience, and the simple mechanics of getting up onto a podium or stage and - in my view, even more important - getting OFF it at the end.

But a whole degree course? I can't envisage a sufficient quantity of material for it.

No. Not enough to it to justify that as an undergraduate major. Beyond one course, it's a matter of experience rather than schooling.