gentleman, on the other hand, is understood to have some intellect, although not necessarily an abundance of it.

I would have said the standard English sense is quite the opposite of this, actually - and that yes, Wooster is the archetypal brainless loon representative of the inbred chinless aristocracy (delightfully harmless for all that!) who is extricated from his own ineptitude by the machinations of the studious Jeeves. Hence the layers of irony implicit, including "you are not only a gentleman, but a scholar (even though you're wealthy enough to pay someone else to do it for you!)"

I'm with you, Father Steve - except I drag a volume from the shelves rather than the wonderful Fry & Laurie interpretations.