This is a great question that comes up every few years, and it's great because it's difficult to settle. My two cents:
If we were to take out the American caveat, it's hard to argue against a Zep or Beatles at the overall top of the heap. So if we think in terms of those guys, 'classic rock', I think the closest American band that is in the same vein with massive staying power and true rock chops is Aerosmith.
If we're going just personal preference for best all around band that was uniquely American and had a lasting legacy, I'll take Tom Petty and the Heartbreakers all day. For those that only know a handful of their (many) hits, if you really love great music I suggest a deeper dive into their 30+ year catalog. Also a killer live band.
Speaking of live bands, Allman Bros are certainly in the running for pure musicianship as best American band.
The Eagles had some great songs, and I do like the Eagles, but for me it's hard to give the title of greatest American rock band to 70s easy listening outfit.
As for the 'new' guys, Foo Fighters have a ton of hits and are really great, maybe just not all that groundbreaking.
Pearl Jam is a decent contender. Some of their very best music was well after their heyday.
RHCP, imo, are the exact opposite. Some great early albums, but just became a parody of themselves after a while. Your songs don't have to be great if they're funky, but I feel like a lot of the funk was gone by the early oughts, and we were left with not a lot of substance. Maybe I'm hung up on the fact that I never understood the fascination with John Frusciante. I just don't get it.
I guess overall though I'd have to go with Hinder.