Which band IS really America's Greatest (rock & roll band)?


When I consider my priorities for this category, I cannot come up with any other than CCR.

Their output as a band was short compared to others, yes..

When I say America's greatest rock & roll band, this = the output or even the basis on which a band formed, had in their DNA, America's roots! It doesn't even matter that we now know CCR formed in California, their DNA as a band transformed their birthplace but it more importantly brought forth the (soul) of get down and dirty) Rock & Roll in it's raw form!

HELL YEAH!
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The Grateful Dead

“They aren’t the best at what they do, they are the only ones that do what they do” Bill Graham

The Grateful Dead, no question. You may not like their music but as far as success, legions of fans following them around the country year after year, the number of concerts performed, years together, earnings, subject matter and lyrics being solely focused on Americana, the old west and the American psychedelic revolution, they’re in a league of their own. We can end the discussion now :) just kidding, of course.
One more vote for NRBQ. And I would add that the current incarnation is also excellent. No qualifications.
@sunnyd, thanks for the reminder. I haven't heard the new NRBQ lineup, of which only pianist Terry Adams remains from the original members. Drummer Tom Ardolino passed away a few years ago, bassist Joey Spampinato (a great one, one of my all-time faves. Keith Richards agrees, hiring him for the band in his Chuck Berry documentary) is in treatment for Cancer, and guitarist Al Anderson moved to Nashville and is now a full time songwriter. My comment above referred to the period when Joey's brother Johnny took Al's place on guitar. Johnny's okay, but he's no Al Anderson!
I love the 19th Century lyrical themes from "The Band".  The historical ballads, the old-time sound, the small-town feel and their forlorn, back-woods style of singing are steeped in tradition, e.g., "Acadian Driftwood", "Cripple Creek", and "The Night....Old Dixie...", come to mind as spiritual and redeeming.  

Similarly, Robert Hunter's lyrics with the Dead often speak of Appalachian folks, oppressed laborers, miners, winos and hoboes--but also of lonely burdens and spiritual rescue. With lyrics such as "I live in a silver mine and I call it Beggar's Tomb"; "Goddamn, well I declare, have you seen the like? Their wall are built of cannonballs, their motto is 'Don't Tread On Me'"; "Tumble down shack in Bigfoot County, snowed so hard, the roof caved in"; "Now I don't know, but I been told, if the horse don't pull you got to carry the load".  And yes, the GD could sing well--better in different periods of their long history.  Listen to "Jack Straw" from Europe '72 for a live example.