Wilko Johnson/Dr Feelgood is a terrific example in my opinion.
David Lowery is a great songwriter and was the force behind Camper Van Beethoven and even better, Cracker.
Stan Ridgway was the big force behind Wall Of Voodoo. He is still putting out great material and tours somewhat in the Western US.
Wow, I’d disagree with that. Alex didn’t have a tenth to ability or innovation of his brother. Most of the first three or four Van Halen albums you could teach a rookie to play in a month. It’s not that hard to play in four with a doubled up bass drum on the three.
And the first instrument I ever played in a band was drums. I wanted to be Neil Peart.
I was quickly informed by experience that I ever would be.
I agree, and don’t see this as contradicting any of the points I made.
How about John Entwhistle? Townshend was a great rock guitar player, and Moon was a great rock drummer, but for my money Entwhistle was a better musician than either.
Too obvious.
Mark Everett in eels
Billy Corgan in smashing pumpkins
Two come to mind for me: Camel and Japan.
The other members of Camel are good - Peter Bardens in particular was a great keyboardist - but not really superlative. Andrew Latimer on the other hand IS a superlative guitarist. Every bit as good as David Gilmour, in my opinion, just more obscure.
Likewise Japan had good personnel - not superlative but good - vocalist David Sylvian is sort of half Bryan Ferry and half David Bowie, very solid vocal performer and creative composer - but bassist Mick Karn was the star of that band. He had a really, really unique style. Any fretless bass virtuoso is inevitably going to be compared to Jaco Pastorius - except Mick Karn. Somehow he took that instrument and carved out a totally distinctive style of his own on it. Karn was a monster. (He died in 2011.)
That’s good, because it wasn’t intended to contradict anything you said. 
My local NPR affiliate plays classical music for much of the day, and one day, I had it on as background noise and the announcer said that the preceding piece was “composed and conducted by Tony Banks.”
Yep. THAT Tony Banks.
John McGeoch would like a word.
Well said!
j
Thought of a really obvious one we all missed:
Eric Clapton while with The Yardbirds
Jeff Beck while with The Yardbirds (but before Jimmy Page joined)
Jimmy Page while with The Yardbirds (after Jeff Beck left)
Beck and Page overlapped for a little while.
Two more:
Drummer Tony Thompson with Chic. Nile Rodgers and Bernard Edwards were both rock solid players (and Midas-touch producers), but Thompson was on another plane.
Duran Duran bassist John Taylor.
While it’s true that John was certainly a bass virtuoso, I’d have to give the edge to Pete in the “great musician” department. After all, a huge percentage of Who songs are covers of Townshend’s demos where he played almost everything.
Alan Gorrie on guitar and bass with Average White Band.
I agree with Wilko. His unique style of finger shredding playing was the sound of Dr. Feelgood.
Yep, Will Farrell is outstanding in the Chili Peppers! 
You have reminded me:
Steve Winwood in The Spencer Davis Group.
It’s a loaded question about musical ability versus ability to play their specific instrument. The latter of which might well exclude singers from the list.
For instance, the main songwriter by a long shot in Pink Floyd was Roger Waters. But reports were he wasn’t the greatest of bass players. But boy could he write and arrange music. Plus the lyrics.
In that band, David Gilmour was the one who could play the instrument of choice as well (up to you if you think Mason and/or Wright are in the running, but to me both disappeared into drugs). Yet he didn’t write very many songs of his own (were many really his, or were they just the solos?) in the main floyd run, and boy he couldn’t write lyrics.
Sometimes the bass players are the musical talent behind the band, without the technical competency they demand from their instruments. Phil Lynott, for instance.
Garth Hudson gave music lessons to the rest of the Band. Taught them all how to play their instruments. Does that count?