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I think twin adoption studies that control for socioeconomic class show heritability cannot be boiled down to environmental stuff the parents do.
That one also always sounded like vodoo to me. I also don't like various statistical tricks they play to normalize IQ, though that one cuts both ways as far as political correctness is concerned.
I think "the general factor g of intelligence" is just "the principle component if you do principle component analysis on things that purport to measure intelligence". Like it definitely exists in the sense that PCA does indeed spit out a large first component if you do that. As long as all of the intelligence tests you're feeding into the statistical process meet some specific criteria (e.g. linearity, similar scale), it very strongly demonstrates that all of the tests are mostly measuring the same thing as each other.
To develop an intuition for what "the PCA spit out a large primary component" means in practice, let's consider an example housing dataset which includes attributes like price, number of beds / baths / parking spaces, presence of AC/heat, etc. If you do a PCA on that dataset, you get a primary component which explains 25% of the variance (and the next components explain 12%, 9%, 9%). Let's call this primary component the "general factor h of house-goodness". Sale price, number of bedrooms, and presence of air conditioning, in particular, are very strongly h-loaded, though every thing that would be nice to have in a house is positively h-loaded.
It's pretty clear that h reflects an actual thing, and that actual thing is probably approximated by "how good the house is". It's sensible talk about things like "high h houses that score low on the price test", or, as we say in plain language "nice houses that are cheap".
What the presence of h does not mean is anything like "this house scores higher on the number of bedrooms test because it has a high h factor".
You should view "the general factor g of intelligence" through that lens.
(cc @RococoBasilica, this seems relevant to your questions)
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So, Wikipedia defines heritability as "a statistic used in the fields of breeding and genetics that estimates the degree of variation in a phenotypic trait in a population that is due to genetic variation between individuals in that population. The concept of heritability can be expressed in the form of the following question: "What is the proportion of the variation in a given trait within a population that is not explained by the environment or random chance?"
But in every single HBD discussion, there is someone claiming that heritability and genetic influence are not identical. They usually bring up something clearly environmental, such as "having books at home" and claim that this would also be heritable. I do think it has something to do with how many environmental circumstances such as parental behaviour might themselves be influenced by the parents' genes. But this doesn't seem like a refutation of genetic impact to me.
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