To get back to the OP, basically it sounds like your friend just likes to argue. If you want to participate, the next time he mentions this ask him what his evidence is that the Beatles did not write their own songs.
Feel free to remind him that “I don’t like them and I don’t think that they could have written those songs” is not evidence.
Paul came pretty close to that on “Got to Get You Into My Life,” and I think there’s another song of Paul’s that’s also largely one chord.
I dunno, Marley23. There’s only one line in the chorus, “Got to get you into my life,” and yep it does (almost) consist of a single note, but the rest of the song, especially the beginning, is practically acrobatic, jumping all over the staff!
Gotta say it’s nice seeing Paul getting his props in the thread. John gets the hipster points and it seems among web discussions he’s usually considered the ‘cooler’ choice of the main songwriting duo. But I think critics and fans who dismiss Paul as the less “deep” or talented of the two, or who consider him just a dude who wrote happy and/or pretty tunes, are mistaken and selling him very, very short.
Natch, the OP’s friend is just plain crazy. 
The vocal melody does, yes, but a lot of that song sits on the G chord.
Well, kind of. Just the beginning part of the verse sits on the G, and then the second half of the verse ("Oh and I suddenly see you) sits on B minor with a bass line moving chromatically down, and then to a IV-II-V-I resolution (“Every single day of my life”). Plus even the ending of each G line has a move to F/G (or you can call it G13). It doesn’t really come close to qualifying as a one-chord song, in my opinion. And then you have the verse, I-IV-V, so you’ve got I-ii-iii-IV-V all adequately represented in the song.
I’ve talked with Rudolph Isley about his English tours with the Beatles in the early 1960’s. He said that John didn’t know much about the technical aspects of music, but if you showed him something once, he knew how to do it. Show it again, and he would improvise and improve on it.
Might an example be his harmonica in (With Love) From Me To You? Harmonica was the first instrument John learned to play but he received tips during 1962 from Delbert McClinton who played harmonica on Hey Baby. I can hear Delbert’s influence and John did seem to improve upon it. At least he put it in the service of better songs.