Are the Beatles the reason why modern music exists

You've only touched the surface with The Beatles. They were of course immensely influential but they were also influenced. And no, it wasn't The Rolling Stones either although they also influenced what has developed into today's "modern music".

You cannot come even close to adequately defining why modern music exists in such a limited context. It is a deeper study than that. A crash course in all this can be had via a visit to the Rock 'n Roll Hall of Fame in Cleveland. There are also many fine books on the subject.

Um, Debussy, Schoenberg, Stravinsky, etc.?

I'm not sure what you mean by "modern music", I wouldn't call something that's 40 years old "modern music", particularly having regard to what's developed since then. However, if you mean "rock and roll" in a generic sense, and its derivatives, offshoots, etc, then I disagree with your statement. It's been around a lot longer than the Beatles or Stones. I think that the end of the big band era after WWII and the early 50's was the period when a newly identifiable musical trend emerged from various sources and influences, and which coalesced into rock and roll, and subsequently, "modern music".

I think that people are imprinted with the music of their youth, particularly in their adolescent years. That is the time which is most important to them musically. My parents think rock and roll started with Elvis. People of my generation think it started with Beatles/Stones. I know a few "younger" people who think Deep Purples' "Smoke on the Water" started modern rock and roll, for no reason other than the fact that it was the first rock and roll they listened to. Ask the great rockers of the 60's what their influences are, and more often than not, the names will be Muddy Waters and other people in the American, black, blues tradition.

By the way, is it just me, or does Bill Haley and the Comet's "Rock around the Clock" from the 50's sound like the "Charleston" and other flapper era music from the 20's and 30's?

Everything was influenced by the Beatles. Yes, everything..

Chris