"Can any other band claim to have better songwriting and/or vocals than the Beatles?"
Context has so much to do with it. Many of those songs, if put side by side with so many contemporary songs, would be found to be somewhat (what's the word?) quaint? Among other not so favourable adjectives. Fact is - if a young up 'n comer dumped even 'Hey Jude' ("don't make it bad, take a sad song, and make it better" - see?) on the desk of some A&R guy today, it wouldn't make it any further than the A&R guy's desk.
You can't argue that the Beatle's weren't extremely talented, but it's next to impossible to do a comparison like the one you're trying do. The actual structure of a great many of their songs is quite basic... but, for example, nobody had ever done such and such a thing before... which in turn makes them "brilliant". It's hard to leave out the phrase "for the time" and it's next to impossible to pretend that the songs aren't by now so steeped in nostalgia, that you could almost say you aren't even hearing the Beatle's songs anymore - you're just hearing the Beatles.