The classism/elitism problem is much more an image problem foisted/reinforced by pop culture than it is a real, inherent issue. When all you hear of classical music is on TV shows where it represents "the rich, elitist villain" or on ads for luxury goods, people reverse the association arrow.
How many people who perceive classical music as elitist actually have a personal experience of someone on the "inside" acting elitist and shunning them as "outsiders"?