But yeah, I think there are problems with assuming that any time you merely think something will happen without knowing, some "magical" element of chance is added to that thing. If this were the case, even math itself could have an element of chance, if you're "unsure about which formula represents the situation accurately" or something. Humans can't turn something into chance merely by acting dumb :)
Math is different because during exam you can double-check your results. I've never failed a math exam and there are thousands or millions of students who never failed. But there is no single chess player who has never lost.
By the way in math your scientific results are scrutinized by your peers and quite often mistakes or gaps are found. But math is a search for truth and such mistakes don't matter for the future. Chess in a game, a competition and your mistakes and losses can't be corrected - they are in databases forever.
"In this case the game ends with the less skilled and less knowledgeable player winning the game."
Yes, but that doesn't mean the game itself was probabilistic. The less skilled player made the better decisions, and had clearer thought, for that game. If we wanted to measure overall skill though, we would want the players to play in various different types of positions and so forth. Besides, a player is responsible for using their knowledge correctly; if they don't that's their own problem.
The game itself wasn't probabilistic: I agree
It's the player's fault for using information incorrectly: I agree
I want to make the claim though that their performance (not chess itself, but their performance) was not based on knowledge or skill and therefore contained some element of chance.