As someone who finds chess problems interesting (I'm bad at them), they're really a third sort of thing. In that good chess problems are rarely taken from live play, they're a specific sort of thing which follows its own logic.

Good ones are never randomly generated, however. Also, the skill doesn't fully transfer in either direction between live play and solving chess problems. Definitely not reconstructing the prior state of the board, since there's nothing there to reconstruct.

So yes, everything Hikaru was saying there makes sense to me, but I don't think your last sentence follows from it. Good chess problems come from good chess problem authors (interestingly this included Vladimir Nabokov), they aren't random, but they rarely come from games, and tickle a different part of the brain from live play.