This video of Oscar Peterson playing Yesterday of Paul MacCartney is an example among dozens of other titles by the Beatles covered by the greatest names of the American jazz scene.
I remember Ella Fitzgerald singing Can't buy me love which has a powerful swing in it. Count Basie also played one title of the Fab Four (although I can't remember which it was), not to forget Ray Charles (Yesterday and Eleanor Rigby) and Franck Sinatra as well, singing Something by George Harrison.
There are certainly many, many other American artists (Elvis, but he may have been a good performer and singer, he's not particularly remembered as a musician) who covered many titles written by Paul and John, simply no name comes to mind at the moment.
This long list in order to make a comparison with the Rolling Stones (yeah, I'm not dealing with metaphysics here) who may be great performers but, IMHO, are poor musicians when it comes to composing.
Apart Ruby Tuesday, Lady Jane and Sitting on a fence which all have tremendous melodic lines, most of the songs by the Stones are shallow musicalywise.
Don't get me wrong here: Honky Tonk Woman, Sympathy for the devil, Jumping Jack Flash, Gimme me shelter and many others are really exciting tunes to listen to but where's the melody?
That may be the reason why the Beatles' songs were so many times covered by American musicians since, for what I know, Americans are impressively good at writing beautiful melodies. Listen to this for example...
Think of Bing Crosby, Irving Berlin, Nat King Cole and so many others.
So sorry but when it comes to compose beautiful melodies (notwithstanding the harmonies and variety of instruments) the Beatles beat the Stones hands down!