So, the only way to counter this strategy is to fill up the board with no one winning, which is a win for white? (LOL, I'm sure you know what I'm trying to say)

Would someone be willing to play out a game of anti with me as black, using the mirror image strategy, and show me how you, as white, get out of it? It will make more sense to me if I am actually playing it out. Unrated/Uncounted, if you prefer.