Nah. Goodhart's law is literally just "if you play a matrix game don't announce your pick in advance". It is not a real law, or not different from common sense. (By matrix game I mean what wiki calls "Normal form game[0]", e.g. rock-paper-scissors or prisoner's dilemma.)
In education, regarding exams, Goodhart's law just means that you should randomize your test questions instead of telling the students the questions before the exam. Have a wide set of questions, randomize them. The only way for students to pass is to learn the material.
A randomized standardized test is not more susceptible to Goodhart's law than a randomized personal test. The latter however has many additional problems.
That's not even remotely true. A randomized standardized test will still have some domain that it chooses its questions from and that domain will be perfectly susceptible to Goodhart's Law. It is already the case that no one is literally teaching "On the SAT you're going to get this problem about triangle similarity and the answer is C." When a fresh batch of students sits down in front of some year's SATs the test is still effectively "randomized" relative to the education they received. But that randomization is relative to a rigid standardized curriculum and the teaching was absolutely Goodhart'd relative to that curriculum.
"The only way for students to pass is to learn the material."
Part of Goodhart's law in this context is precisely that it overdetermines "the material" and there is no way around this.
I wish Goodhart's law was as easy to dodge as you think it is, but it isn't.
I do not believe schooling is purely an exercise in knowledge transfer, especially grade school.
School needs to provide opportunities to practice applying important skills like empathy, tenacity, self-regulation, creativity, patience, collaboration, critical thinking, and others that cannot be assessed using a multiple choice quiz taken in silence. When funding is tied to performance on trivia, all of the above suffers.
In education, regarding exams, Goodhart's law just means that you should randomize your test questions instead of telling the students the questions before the exam. Have a wide set of questions, randomize them. The only way for students to pass is to learn the material.
A randomized standardized test is not more susceptible to Goodhart's law than a randomized personal test. The latter however has many additional problems.
[0] : https://en.wikipedia.org/wiki/Normal-form_game