I don't know how you are defining randomness, but the best player doesn't win the game every time, which is about as much proof as you need that there is a degree of randomness.

Define random in terms of sports, the same conditions apply to chess. The wind affecting trajectory of a ball could be similar to environmental factors effecting cognitive performance.

I wouldn't use the term "random" either. What randomness exists for baseball? Slight curve of a bat? Maybe that is part of the game and should be calculated into by the player and isn't randomness at all but part of the game. "Randomness" exists in almost every competitive activity, its called variance.

It's not even worth discussing, you don't even understand the terms you are using.