To recap, my position is this:
You talk about probabilities and averages. Probabilities just describe the problem, the conditions of the wheel. Probabilities is another way to say "Well, I don't know for sure"
(because a single number can come before Red in your example)
I'm talking about the real outcomes. I'm talking about the prediction of outcomes, of any outcome at any time, with absolute certainty.
So, in my book, no matter the underlying probabilities, a game where you cannot make a single 100% certain prediction, in which every spin outcome is unpredictable, is a random game.