Even with random moves, White is going to win more than 50% of the games. Who knows? White might even win 60% of the time!
Now imagine a series of games played between Kasparov and Karpov, and in every game Black gives Queen odds. Do you think that Black could even manage to draw one game in a thousand?
Evidently, stronger players need less of an advantage to force a decisive result. (Duh. What a revelation!)
The converse should be true as well: if a pair of players plays a series of games where they give each other Knight odds, the average rating of those two players could theoretically be derived from the winning percentage. In practice we do not have the data that would tell us "if the odds-giver won one game in five, the average rating of the two players is 1600", or something like that; but if we did, it would be possible to give people (or new computers) fairly accurate ratings without their having any exposure to the general public of chessplayers.
Because small differences are more significant to stronger players, a game of Chess with Different Armies could be very equal for masters and very unequal for grandmasters.