But when the weaknesses it has require looking 20 ply into the game, can anyone exploit those weaknesses? And furthermore, if the computer itself is able to see 20 ply into the game, then it can spot its own weaknesses and you need to look even further, making the question of whether it's really a weakness.

Weaknesses are only relative to capabilities of the opponent to exploit them. If a tank has a weak spot that rockets can hit, but it's being opposed by humans on horseback, is it really a weakness in that context?