Actually, I can't believe that no one's mentioned the Grateful Dead as one of the seminal American bands. They count as being excellent musicians and pace setters of popular culture. I think fans would consider them wholly as a band and not just Jerry Garcia's front organization. (And wouldn't "Cherry Garcia" count as the first ice cream inspired by a band?)
Their studio albums never manage to capture the magic of their live performances and recordings but they still manage to score hits across the decades:
Truckin'
Casey Jones
Uncle John's Band (My personal favorite Dead song)
Touch of Grey
Good stuff. (And no. I never managed to catch them live.)
ETA: I would consider the Byrds an excellent band of their time. I think their songs stand the test of time, but as a band they didn't have the longevity that I think would mark a really great band. (I should disclose that
Eight Miles High would probably be included on some sort of desert discs package, were I to draft one.)
But they get extra props for the popularization of the 12-string guitar. And for inspiring George Harrison. And covering Bob Dylan so well. (I like their version of
My Back Pages, even though it is tied as my favorite Dylan song, period.)