Well, in a way, these may be no-brainers. It usually worked out ideally that the Beatle who sung the song that he wrote probably sung it best. Some songs that John wrote demand a John vocal,and the same with Paul. Some could go either way.
Golden Slumbers, Hey Jude. John couldn't have touched them.
Lady Madonna, Helter Skelter could have been a toss up. I dare say I can even hear John singing Yesterday, but he probably wouldn't have (except for the hilarious Fan Club Christmas record version where they all sing together out of tune.)
I think John did Baby You're A Rich Man, And Your Bird Can Sing, In My Life and Girl better than Paul could have.
I'm sure there are other examples.
A Day In The Life is perfect as it is.
I think John dominated vocally in the earlier Beatles, but no question, Paul came into his own. I could never say, though, that one was generally better than the other, they both were gigantic equals.
As far as harmonies go, the Beatle blend was so right, nobody came close except maybe the Beach Boys and C,S,N & Y. But I think they only got close, the Beatles were still way ahead. That was the main thing missing for me when the Beatles broke up, the harmonies.