>(You could of course argue that they did not give enough context ... but that's exactly my point: the trick makes the proof work but hides the explanation)

Can't one see it in another way: that the trick illuminates a deeper explanation, connection the theorem's context and the stuff that's conceptually outside of that context. And that the problem is we don't know why the two domains (the context and the conceptually outside of it one) are related and cooperating in this way.