That's an interesting notion that may prove true in the future, but a few exceptions aside, I don't think it accurately describes the history of what you'd call "the western canon." The reality is that in the past, most popular art has been completely forgotten a few generations out. Of course there are exceptions, but the overwhelming number of plays, songs, novels, paintings, etc are dust. Your example of Shakespeare was a bit off the mark, because while he certainly had and has popular appeal, Shakespeare was in no way 'basically popular entertainment'; Lord Chamberlain's Men, The Kings Men—they were not performing for the sole benefit of the groundlings, they were performing primarily for very wealthy patrons.
Also, Shakespeare created perhaps the finest and most elevated art in the English language, and was recognized for it then, even more so today; the plays that were basically popular entertainment in his time are footnotes or worse today.
And as a side note, I'm one of those people who enjoys seeing Shakespeare performed. I think you'd be surprised how much audiences enjoy it, and not just in that insufferable, look-how-cultured-I-am, self-congratulatory way.
Music's another good example--from any objective standpoint, there's music being produced today that matches or exceeds the artistic merit of any of the old composers; likewise, tons of the old composers were popular musicians in their own day. But there's an awkward separation between high culture and popular culture, mostly based on how old something is.
It might come off as silly to actually portray this in science fiction, but it's interesting to wonder how people a century or two from now will approach the art we're producing today. Will there be cover bands performing Metallica's "Master of Puppets" who are treated with the same reverence as contemporary orchestras performing Wagner operas? Will people wear formal clothing and sit quietly, not realizing that it was meant to be music for sweaty teenagers to run around and body-slam each other to?