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? Is it inevitable, or is it the result of methods of education?

Some degree of this is probably inevitable. Integration in particular has no closed solution (unlike differentiation), so there really is no one general method you can apply to all problems. All you can do is remember a bag of tricks. While for differentiation, a few general rules allow you to integrate all elementary and trigonometric functions, and that's pretty much all you encounter in school.