If anything, I think the AMC exams are more vulnerable to prep than the SAT/ACT in the following sense: I think a bright and disciplined kid who does lots of self-study (e.g. using the free Khan Academy SAT materials [1]) has a decent chance to do well on the SAT/ACT, even compared to a bright and disciplined kid whose parents pay for classes. I think the gap is bigger for AMC stuff because, IMO, it's harder to teach yourself math competition skills. In that sense, access to paid instruction becomes a much bigger edge.

Math competition skills also test something very specific and are a pretty useless metric for most people.

[1] https://www.khanacademy.org/sat

Good luck self studying for those -- I tried. I made AIME but had no chance there.

There's no way a significant number of generic public high schools start caring about the AMC. It will just be helicopter parents, wealthy private schools and the odd self motivated student.