You left one type of person out...
Kinks fans
We weirdos like like the Kinks better than either.
I like The Beatles and Stones almost evenly for different reasons. The Stones made music I like more and I think it's better rock, The Beatles made better pop and were lyrically heads and shoulders above the The Stones.
Then you have The Kinks - lyrically superior to The Beatles and rocked as hard as The Stones (when they wanted to). The nasal voices and bad luck kept them from being as loved as they should have been.