Classical compositions appeal at multiple levels -- they have to, because composers at that time were, in large part, composing music for the public. They had to make a visceral connection. That’s why even the non-classical listener knows the tunes to many compositions.
Classical composers - the vast majority of them, anyway - made music at the pleasure of European monarchies, courts and assorted hangers-on. The public was not concerned.
This built-in perception of elitism endures. While most folks may know the first 20 seconds of Beethoven’s Fifth because it is incontournable, classical is perhaps the only music genre that periodically faces calls to be either removed from the curricula of public universities or to have public subsidy of its study and performance curtailed or ended.