J.B.
The "multiple of 12" comes from the fact that kids at this grade level have already mastered or are mastering their multiplication facts up to the twelves. So to me it's simpler than you stated - there are 2 twelves in 24, so 6 times 12 is the same as 3 times 24, which is 72.
It's not about just solving problems using rote memorization or by using long multiplication, which is what 24 x 3 would require at this age. Yes you're right in that the goal is to do more work - that way they'll learn greater flexibility of thinking, which is good not only in higher level math but also translates well into all other areas of academia. One of the coolest parts about teaching for me (I teach test prep to high schools part-time) is that when we're working through math problems, the kids will come up with all different ways of solving, including some strategies that I hadn't considered. Some are clunky and cumbersome, but some are very elegant and efficient.
I know it can seem ridiculous and tiresome at times to learn things using every method but the most obvious, but there really is a method to the madness. Just try to go along with it.