A start in that direction is kind of seen in AI Sensei’s Training Mode feature
As far as I know, you can make puzzles out of your mistakes in a game and then do a quiz on them later. I think at the moment it’s a bit more like find the right move, as opposed to solve the puzzle completely and the answer might depend on the level of support you pay for the site. That is, if the AI does more playouts in the game review it could favor different outcomes.
Then a recent version of Katrain has this local positional analysis mode
I wonder if there could be some sort of automated system between the two, where you give the new program an sgf file, it reviews to find the biggest X number of mistakes. Then it uses a local analysis mode to decide if the mistake is a local one - that is, was the best move also local as opposed to a tenuki. If the score loss is big enough, it could be that something got captured, or some endgame went wrong etc and it might be useful as a puzzle. Then save the local variations as a tree to make a puzzle from.
It could be a bunch of work, but it wouldn’t involve training a new AI on generating puzzles at least, that and even if the AI could generate the puzzles, it should also be able to solve them 
Maybe @sanderl has thought about or knows about projects like this before?