I don't think the Stones are "generic", just thinking through their catalog: they have light little pop tunes ("She's a Rainbow"), the have lots of blues both Delta blues songs("Love in Vain") and Chicago Blues ("Little Red Rooster") and their own blues-flavored originals ("Sister Morphine"). They have country songs (or mock-country, perhaps) like "Girl with the Far Away Eyes". They have a lot of rockers: "Start me UP" "Brown Sugar" "It's Only Rock and Roll". They had a disco period: "Hot Stuff". Latin Jazz flavored jams "Can't You Hear Me Talking".
They were one of the original British Invasion bands, so they invented a lot of what we consider rock & roll after about 1969.
I don't quite see it as "generic".
They were pretty generic blues cover band in the 70’s. I like their stuff, but very little of it was more than a sugar high. It was fun, but forgettable.
But that applied to most of the 1970’s music.