In a game design context, he is definitely using "uncertainty" in a wider sense, as popularized by Greg Costikyan's Uncertainty in Games book.
In that sense of the word, it's not only about random things, but also things like "will I click at just the right time to head-shot that enemy?" or "I will checkmate the next turn unless my opponent thinks of some clever move that I don't?"). And the theory is that once you run out of uncertain things there is no more a game, as the player know how it will end and there is nothing more that can fail or anything unexpected that can happen. Basically like reading the end of a book you have already read before, so you know exactly what will happen.
And depth from a game design pov is also not necessarily strictly positive. Make the game too deep and there is, as you say, pure random. You could keep adding rules to chess to make it 100% impossible for any human to remotely guess what kind of move to make, and that's when you added so much uncertainty that it became too deep.
In that sense of the word, it's not only about random things, but also things like "will I click at just the right time to head-shot that enemy?" or "I will checkmate the next turn unless my opponent thinks of some clever move that I don't?"). And the theory is that once you run out of uncertain things there is no more a game, as the player know how it will end and there is nothing more that can fail or anything unexpected that can happen. Basically like reading the end of a book you have already read before, so you know exactly what will happen.
And depth from a game design pov is also not necessarily strictly positive. Make the game too deep and there is, as you say, pure random. You could keep adding rules to chess to make it 100% impossible for any human to remotely guess what kind of move to make, and that's when you added so much uncertainty that it became too deep.