One clever way this is measured is twin studies, which also are not what most people, particularly those who prefer to write more than read, think. You don't search for twins separated at birth, but instead compare the differences in a trait between identical and non-identical twins. If the variation is greater, then the trait is generally significantly heritable. So for example - height would be an obvious one. By contrast the variation in accent between identical and non-identical twins would be zero.
Direct genetic causality is not the only mechanism through which genes select for phenotypical traits. Genes also select and interact with the environment.
Unlike the individuals you have cited, James Watson is a geneticist, spent his entire life studying and working on genetics, and in fact was even the person who discovered the structure of DNA. But because of his views on the genetic aspects of IQ (which inherently becomes intertwined into race, as race is just shared genetic ancestry), he was completely demonized, his career destroyed, and various honors revoked. Higher profile people speaking on these topics publicly know this all too well, so it mostly just turns into cheap virtue signaling as opposed to adding some genuine insight.
In your case, the examples they've offered are simply wrong, as would be immediately apparent with the most typical method of measuring heritability!
This is a "not even wrong" situation. Is cognitive ability significantly genetically determined? Maybe, maybe not. A broad heritability statistic from a twin study isn't going to resolve the question.
Here's a good link for you:
http://bactra.org/weblog/520.html
I promise, the author has studied and thought more carefully about the question than we have.
Fair warning: you would not be happier if I cited a molecular geneticist on this subject. Your argument gets even harder to sustain once you bring GWAS into the picture.
It'd be akin to arguing to somebody who wants to claim the Moon landing was faked, and after the rather straight forward rebuttal of their argument links to some blog in the tens of thousands of words from some statistician they claim is "very smart." It's silly.