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.