Am I misunderstanding something or is there a glaring statistical error here? I think this analysis works if the matches are all between two players of truly identical skill that will roughly win 50% of the time against the other, but that's not really the case here. They attempt to do that by controlling for Elo but I'm decently sure that you could pick any two players of the same Elo on lichess and one will win the majority of games.

Now if we assume there is often a slightly better player in these games, the better player will more likely get into an advantageous position early in the game and win more often, but not entirely because they got into the more advantageous position. What I'm trying to say is that someone who is up a rook will win partially because they're up a rook and partially because they are likely a better player in the first place and will continue to play better than their opponent.

I think to do this study correctly you'd need to place new players in a random position drawn from the dataset and actually evaluate the win rate without the confounding factor of having gotten into that position in the first place.