I interpreted the article as making exactly the opposite of your point: because Khan Academy is limited to automated grading, they can only ask standardized-test style questions: solve an equation, select a multiple-choice answer, etc. Meanwhile, the Common Core standards emphasize argumentation (i.e. proofs), formulating equations, and other high-level skills that are harder to test directly but are much more important to actually getting a deep understanding.

Common Core is not a set of tests; it's just a curriculum that sets some basic standards for what students ought to learn. It's reasonable to be concerned about standardized testing as a means for assessing whether those standards are being met, but it doesn't sound like you actually have any objection to the Common Core curriculum.