I'm not sure I agree. I've seen lots of people in museums, the vast majority of them walk around, look at a work of art for 10 seconds and move on. Not that there's anything wrong with that, but my point is that they're appreciating it. They look and decide if they like it or not, move to the next one.
The kind of deeper questioning you describe is rare I think (and to your credit). It doesn't require an education, but that does make it easier.
Incidentally, the average person today has been exposed to more art than most people in history, simply through television and architecture. Moreover, the innovations from the masters have been assimilated in later works until they are now commonplace. That makes it difficult for most people to recognize the novelty in masterworks. They just don't look as impressive when viewed from the future.