For us, writing was the same way as yours ("strongly advised to start in certain ways and it's best to just memorise the beginning off examples and just change few words"), but it only happened a few times at most, and more towards the latter years, while reciting stuff in lit classes was the norm the entire time.
And now I am starting to recall another type of assignments going until late middle school, where we had to write passages from textbooks in cursive in our "homework notebooks", mostly with small changes. Like "here is this 1000 word passage in the textbook written from the first perspective, rewrite it from the third perspective in pen". Made a typo? Well, restart, because while a couple of edits won't take much off your final score, any more than that makes a full restart a more worthy option (because it was required to use pen for those instead of a pencil; more diligent students who didn't wanna bet on getting it right on the first try, they did it first with a pencil and then traced it and erased the pencil). Another evening spent rewriting the same dull passage multiple times by hand in pen.
For the record - my handwriting sucks.
I agree, however, doing it way past elementary school seems like a solid way of wasting time. When you are trying to learn writing or you are in elementary school, sure. But spending hours upon hours rewriting long passages multiple times due to random non-editable typos in late middle school felt mind-numbing and downright awful.
And of course, as you progressed in grades, less attention of graders was focused on the actual handwriting quality, with passages getting longer and more convoluted, so the handwriting tended to degrade the further you got in your elementary/middle/high school education. Not even mentioning what happened to it after high school, because by then (unless your handwriting was completely unreadable) no one cared.
And no, it didn't help in the long-term with handwriting at all. None of the adults tend to have textbook-good handwriting, it would barely even get a passing score in the best scenario (and that would be an exception). And just like in western countries, let's not even mention doctors' handwriting (but that's completely irrelevant to my point).
I've been in several provincial schools in 1990s, good and bad, and we've always had strong anti-memorizing sentiment in math classes, despite in literature classes memorizing was mandatory.
Both make perfect sense.
Now I heard it's turning more to the Western model, though.
I find it hard to believe, given my personal experiences and the fact that having to memorize a poem or a short story and then having to get up in front of the class to recite it word for word for a grade was an extremely common recurring homework assignment in literature classes in my Russian school (from elementary to high school).
Extra details about that type of a homework assignment for those curious: they cannot call up every student due to time constraints for each class period, so for every such assignment, only about half the students get called up (for some specific works that are "more important", they might call up everyone, but over 2 class periods; that was extremely rare though). But those assignments were so ubiquitous, you essentially got called up to the whiteboard to recite at least once every week or two.