My vote, based on the extensive criteria laid forth in the OP, is The Who. The Rolling Stones, in my unpopular opinion, is a pretty unessential group; any of the other three are deserving of a vote based on something.
I’ll vote the Beatles, for the same reason as the other Beatles voters, their diversity. This is especially important for a radio station playing their songs all day: No matter how much you like a given style, it’s going to get old after a couple of hours.
I know and like pretty much everything the Beatles put out. Same with Led Zep, but the quantity’s not there. I don’t much like the Who, and have never desired to buy any of their albums. I really like a lot of the Stones’ songs, but they have put out scads of albums from each of which I have only heard a couple of tracks. If I wanted to investigate any of these four bands’ music farther, I’d pick the Rolling Stones. I probably won’t, since I just don’t care much about pop music any more.
Anyway, going with all that, I’ll pick the Beatles.
Stones, but it was a close call between them and the Who.
Although I like individual songs, The Beatles never really grabbed me. They mostly lacked the harder edge that the other groups had.
Led Zeppelin, because I couldn’t decide and they were first on the list.
The Who.
The Beatles, of course.
I wouldn’t vote for the Stones if they were the last band on earth. Which they may be, someday.
Another vote for Led Zeppelin.
Hmm,
Zep: Jimmy Page
The Who: Pete Townsend
The Beatles: George Harrison, John Lennon, (and sometimes) Paul McCartney.
The Rolling Stones: Keith Richards, Brian Jones, Mick Taylor, Ron Wood.
Really, if it were just Keef and Mick Taylor, I’d probably have to pick the Stones. Mick’s the best of any of them. Add Jones and Wood to the mix, and my only acceptable choice is clear. Stones, every day and twice on Sunday.
I voted for the Beatles because I figure they were like the Beatles of rock and roll.
The Stones were essential in bringing the blues back to the (albeit white) USA. I don’t think The Yardbirds, Animals, Paul Butterfield BB, et. al. would have had the same overall effect. Plus, no Stones, and I don’t see Zep even being a reality.
Not to hijack my own thread…thanks, everyone - no one said this was gonna be easy 
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Y’unnerstan’, if someone asks me to pick between the Beatles and the Stones, I say, “the Who?”
I had to think what my snarky answer would be to dodge the four of these. Well, Floyd, yeah.
But it’s the Who, as if there was even a contest.
What now? :dubious:
I don’t remember WNEW doing that, it sounds more like a WPLJ thing.
I voted Zep but could have as easily voted Beatles. I love both along with Pink Floyd & Yes as my top 4. The Who are my next tier with Allman Brothers & CSN&Y.
The Stones are not even top 20 for me and might be far lower. I do like their early stuff but after 73 or so I can’t stand their music.
I love most of the Zep catalog so I will give them a slight edge over the Beatles who have to have as many songs that I love but more that I only like.
Kind of a tough call for me but I went with Led Zeppelin. As I recall I didn’t really like them (or really know much about them) in the '70s (gimme a break - I hadn’t even hit puberty yet) but the more I’ve heard of them over the years the more I like them. That’s not to say that I don’t like those other three bands, but if you’re asking me to choose one I’d have to go with Led Zep (some of my favorite songs of theirs: “When The Levee Breaks” [okay, that wasn’t really theirs, but they covered it in a whole new way], “Kashmir,” “The Rover,” “Hey, Hey What Can I Do,” “Trampled Under Foot” - there are more but I don’t want the list to go on forever :p)
I love all four, but the Beatles had the most variety overall.
Won’t get fooled again.
The Who are the most consistently powerful-thinking of songs like See Me Feel Me, Who Are You?, or Won’t Get Fooled Again. Zep had their moments, but more often than not just spun their wheels musically. I got weaned on the Beatles and have quite a bit of their stuff, but for them it’s more like quiet admiration at a distance.
The Last Time is the only Stones song I have, and that is unlikely to change anytime soon.
A while back I was doing inventory of my music, and realized I had nothing of the Stones. So I got that big “40 Licks” collection. There wasn’t one song that I’d ever listen to again. So I put them in my 28,000-track iTunes collection that plays randomly. That’s the only way I’ll ever hear them again, and that’ll be too often.
I guess I would put it this way:
If you asked the Stones, they would have to say the Beatles - Mick and Keith getting humbled by John and Paul finishing I Wanna Be Your Man inspired them to start songwriting, fercrissake.
If you asked The Who, they would have to say The Stones - Pete’s #1 idol was Keith Richards - he copped his windmill move after watching Keith warming up at a gig. And since the Stones would cite the Beatles, QED
If you asked Zep - they better fuckin’ say a combo of all three 
- Jimmy Page tried to evolve the Yardbirds into Zep by introducing Keith Moon and John Entwisle into the group (see Beck’s Bolero) then you have to go with them saying The Who, which points to the Stones, which points to The Beatles.
So - The Beatles. But I woulda picked 'em anyway