I too heard the Stones version first. That whole Stones debut album was actually quite good and has aged well.
Yes, they were a blues cover band, but with a style of their own. Jagger didn't sound white, yet he didn't really ape black singers either.
There was only one self-penned song on that debut, but I thought that great as well, very catchy. It was basically a doo-wop pastiche.
Re Rufus, a black artist with a white backing band, was that a usual sight in 1964, Dave? Or was that the house band of the show that simply backed everybody?