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I mean I can try, but I already wrote an explanation in my blog post, so I'm not sure how much more I can say about it. Is there some specific part of my linked post that you find confusing?
I've seen HBDers point out that there is a genetic correlation between homosexuality and mental illness, and use this as an argument that homosexuality and mental illness are innately related, as a counterargument against e.g. homosexuals being bullied and becoming mentally ill as a result of that. (I've also seen lots of other examples, but this was one of the key examples that made me decide to write the post.)
If a genetic correlation referred to a genetic confounding aka horizontal pleiotropy aka "one gene has two unrelated effects", then that counterargument would make sense. The bullying theory of gay mental illness doesn't predict that there is a biological connection between homosexuality and mental illness.
However, genetic correlation actually refers to something more subtle. Two variables are genetically correlated if genetic factors that contribute to one variable also contribute to the other variable. So for instance, intelligence and education are genetically correlated, because genes that contribute intelligence makes people better able to pass exams etc., which unlocks better educational opportunities. In this case, intelligence and education are not genetically confounded; rather they are just ordinarily causally related, and this makes them genetically correlated, just as it makes them ordinarily correlated.
Your example seems pretty easy to test. There are lots of twin pairs where one is homosexual and one isn't the heritability is only 30%. You could just see if the non-homosexual identical twins have the same rates of mental illness as the homosexual ones.
Yes, this is called environmental correlations, it is correct that the phenotypic null hypothesis also predicts the environmental correlation to be high whereas the genetic confounding hypothesis doesn't predict that. (Specifically, the phenotypic null hypothesis predicts every variance component to be correlated, whereas confounding hypotheses only predicts confounded variance components to be correlated.)
Some of the studies on homosexuality and mental illness finds the environmental correlation to be zero, which supports the genetic confounding view. I have at times acknowledged that/pointed that out. I also address the concept in my linked post.
The thing that bothers me is not the conclusion but instead the argument: WHY would HBDers make the argument with genetic correlations in the first place, when clearly it is the environmental correlations that are the key question? Because they don't know the phenotypic null hypothesis. But WHY would HBDers not know the phenotypic null hypothesis when it is such a basic concept for heritability? Because the phenotypic null hypothesis is anmoying and sounds like an outgroup thing, is my hypothesis.
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So, then just use how the correlation between being gay and mental illness varies over time or between countries with different levels of homophobia. We know from IQ studies that heritability is very stable over time, between social classes, between races. etc. The onus is one the people who imagine some kind of fanciful complex gene-environmental effect to prove it, rather than the simple gene x causes y about the same in almost all environments a normal within 2 standard deviations person would encounter.
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