The first paragraph of that Wikipedia article affirms my opinion:
> The correlation between IQ scores and grades is about .50. This means that the explained variance is 25%. Achieving good grades depends on many factors other than IQ, such as "persistence, interest in school, and willingness to study" (p. 81).
While that paragraph specifically mentions school grades instead a more narrow subset of certain specific tests the point still stands as affirmed by the APA. The numbers you quoted from the third paragraph allow are only about 80% aligned. That is a tiny bit better than half between perfection and a random guess.
The unfortunate reality is that performance on those tests is primarily a direct correlation of time and money spent on preparation specifically for those tests. A major differentiator between those standardized tests and many general IQ tests is centrality versus assessment distribution. You wouldn't for example test creativity with a multiple choice question with only one correct answer the same for everybody.
Look, most people who are from a higher income population want to believe that standardized tests are a fair reflection of intelligence, because with enough practice and preparation they can score higher and thus believe they are more intelligent than the population at large. If the population at large spent less effort preparing for the test then its value as a tool of performance compared to other people is less valid. That is selection bias.