Great points. That wasn't quite what I was shooting for. As a newbie to thinking about this in detail what jumps out is how well a similar set of techniques does generalise to an approximation of the right behaviour. It is still an approximation as you say.
The fun thing for me about the Deepmind paper is that they do use Stockfish to bootstrap a general approach from a specialist one. That is exciting - why is the but required in your post?