As much as I hate to break it to you, I can't imagine people liking Schoenberg, Webern or Stravinsky (no, I don't even dare mention Varèse) more than Mozart, Chopin or Bach. In fact, just from my accidental observations, I would imagine this as something that people might like:
People are scared when they hear weird things, sadly. It must be quite consonant and smooth for them to like it (OK I'm not saying the Liszt piece doesn't have chromaticism in it, but it's all kind of smooth, you know). I think the problem lies elsewhere. Perhaps in those labels given to it ''classical music is music for old people and snobs'', ''classical music is dead'' and so on. The author of the article that Morty posted calls the term 'classical music' a 'masterpiece of negative publicity'. There might be something to it.
Part of the problem could be music education (or lack thereof) at public schools. In my country, kids aged 6 to 14 or so (ie. 1st to 8th grade) do have a subject called 'music education'. But this is another masterpiece of negative publicity. The first 5 or 6 years they just sing Slovak folk songs, and then there is some very brief introduction to art music. I know that there is no mention of jazz and other genres that emerged in the 20th century, which is a big problem. I believe the main goal of public schools should be to let kids know that other alternatives to radios and charts exist. But they fail to do so and they even do the contrary.
What about education in your countries?