I am not sure that a classic is infallible in its ability to move you

Ah yes, there is the rub, shanks!

My mentioning appropriate context was sufficiently wooly, but...

I think you can only talk of a 'classic' within a particular cultural context. If you've had no exposure to that culture, and the culture is sufficiently dissimilar to your own, it's difficult to appreciate the work, and it may fail to move you. However, you can acquire a taste for diffent cultures, and you can learn how to appreciate their works. Something that starts out fairly intellectual (though usually driven by a genuine enthusiasm and willingness to learn) can evolve into genuine appreciation.

Most people within a given culture will be moved by that culture's classics. But not all of them will be moved by all the classics all the time - that's just human nature.

There's some kind of relation to teaching classics here. If I hadn't been more or less forced to learn Shakespeare at school, would I appreciate the Bard's works as much as I do now? I'd definitely never have made the effort to read Chaucer, and that would have been a loss. I'm very grateful now for almost everything I disliked learning at school, much as I hate to admit it!

Learning appreciation, acquiring tastes, hmmm.

Yep, round and round....