I've been a Dowland fan for probably 40 years now. At some point after my own discovery of him, Sting recorded an album of his songs. I wasn't surprised at all. It's not an accident that he recorded Dowland, rather than Bach, etc.
I am a major Beethoven fan and regularly listen to him (my most-listened-to Beethoven piece is the Hammerklavier sonata, which I listen to in several different performances with great attention to the differences). I'm not remotely ignorant of what somebody like Beethoven brought to the table.
But the Beatles were much more influenced by the popular songwriting tradition going back to, and through, Dowland, combined with American blues, than they were by Beethoven, Mozart, Bach, etc. Even implicitly.
The Beatles had a huge influence on popular music. First from a music theoretic standpoint, particularly with their use of flat-seventh chords as if they were "normal" chords. Second from their bringing non-rock traditions into the rock world, such as Indian music. Their effect was huge. As Dylan said about his experience of their breakout songs such as I Want To Hold Your Hand, they were “doing things no one else was doing. I knew they were pointing in the direction that music had to go.” He concluded, “It seemed to me a definite line was being drawn. This was something that never happened before.”
I won't debate this any more, life's just too short, but I really suggest you deeply listen to Dowland, and then add the American blues to that, and then the harmonic and stylistic innovations of the Beatles, and consider specifically Bach had to add that had a larger effect on popular music.
I love the Beatles, but I'd _strongly_ disagree that Bach (and later Mozart or Beethoven) had less influence on popular music.