>I think what might actually be hard to game is teacher recommendations. After all these people have been with the kid for a long time and know what they can do. They also tend to be distanced enough that they don't have to say good things about every kid.

Recommendation letters aren't worth a lot. Most kids who have a shot at getting into a top-ranked school surely have some teachers they got along with. But then there's the luck of the draw whether said teacher will also put in an effort beyond a pro-forma recommendation.