Hope this isn't too off-topic, but I wonder if you have any ideas about why that is.

I think that the point is that more people are capable of routine tasks than of conceptual understanding, and that educational institutions want lots of people to do well in math class on account of a desire for (the appearance of) egalitarianism.

In the far-distant past, when I was in school, learning elementary calculus meant rote drilling on techniques of solving integrals. Is this still the case?

What time period was this? (No need to answer if you'd prefer not to :-) )

Math, more than any subject I know of, seems to require a very high level of sheer diligence to get to the point where you can start thinking about it deeply. Is this inevitable?

Some diligence is necessary, but not as much as it appears based on standard pedagogy. I wish that I could substantiate this in a few lines. If you say something about what math you know/remember, I might be able to point you to some helpful references.