While I appreciate the (very good) tutorials about how the transformations work, I don’t think any of these have really answered what I see as the heart of skraboutcha’s original question: why use these counterintuitive transforms instead of something conceptually simpler, like including location and rotation input value for each drawing command?
The closest to an answer that I saw was at the very end of glv’s last link, which has 2 quick videos with titles like “The 12 Rules of Animation”, which include things like the ability to stretch, squash, and move along an arc as being desirable traits for animation. Perhaps these transforms, in the hands of skilled practitioners, make such animation techniques easier than more straightforward position/rotation would. That is the sort of insight I was hoping that answers to this question might give.
Because, skraboutcha, I agree with your premise - that this transformation approach seems frustratingly convoluted, and I have to keep writing code to ‘undo’ it (like using push and pop) so that I can get my images to look the way I intend. I bet there’s a good reason for it, I just don’t see it (yet).