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.
[0] : https://en.wikipedia.org/wiki/Normal-form_game