I think this actually hits the nail on the head. Chess is very much a game of chance.
People have been saying "the players are nondeterministic not the game," but epistemically what is a game with no players?
I play chess seriously in a tournament setting. While it is true the board is revealed, the board is not the game. The board is the operating theater of a game which is entirely within both players minds. People have become overly dismissive with the idea that the game is deterministic because chess engines exist. But chess engines are not even necessarily deterministic because the result of the alpha beta search still vary based on depth and the evaluation heuristic.
Without an evaluation function you cannot even interpret a board position at all. This function is simply not deterministic (unless formalized and then it will be imperfect to situation), and any GM will tell you that some positions are impossible to evaluate.
Additionally there are many times where engines produce suboptimal results until the search space is collapsed by a player like Hikaru. It's not frequent but it definitely happens.
Chess is a game between two Turing machines sharing a tape. Sharing a tape does not allow you to see the state of the machine in each players head. And if you have no players you have no game. This makes chess highly probabilistic in so many senses that using the game theoretic construct of "prefect information game" causes game theorists to seemingly literally not understand it.
Chess is a deeply human game subject to human variance. This is nondeterministic. Poker is also human, in other ways. Anyone who sees chess as deterministic is probably a weak player (less than 2000 fide) and doesn't understand this fundamental aspect of the game. At the higher levels there is even forms of bluffing and swindling, which is fantastic simply because you CAN see the same board and you were still able to manipulate the other player with their own prejudices.