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As it happens, I often run into trouble due to my conceptualist views on the foundations of mathematics (that is, I'm a hardcore physicalist & anti-Platonist cognitivist, which is quite rare among mathematicians), so I find your criticism particularly unfair.
The truth is, for these kinds of studies, the whole enterprise might as well be metaphysical; people saying these kinds of things have formed a religious conviction about the heritability of behavioral traits, and their real objection is that science continues to be done on the topic at all. Ironic, given the frequency with which they complain that this science is suppressed.
(I too have a near-religious conviction about this subject, though in a different direction; I do not, however, pretend that conviction is itself a methodological critique!)
Whether or not you like MISTRA, “they left out the non-identical twins” is a side issue, and the broader evidence that IQ is substantially heritable is extremely strong.
Even if DZA were excluded, MZA alone provides a solid heritability estimate. The DZA sample was small and noisy, and MISTRA uses other twin and family data. Plus, meta-analyses confirm high IQ heritability, typically 0.5-0.8; Later structural-equation models applied to the full MISTRA cognitive dataset (MZA + DZA) estimated the heritability of general intelligence around 0.77, essentially the same ballpark as the original simple estimate. laplab.ucsd.edu. Intelligence is highly heritable, potentially reaching 80% in adulthood, supported by further studies like the Haworth et al. meta-analysis, showing age-related increases in heritability.
Strong evidence for polygenic and SNP heritability is shown from Plomin & von Stumm's 2018 research, showing how polygenic scores predict general intelligence.
Adoption studies further support the genetic influence, as adoptive siblings show weaker correlations compared to biological ones.
Environment is certainly NOT outweighed entirely or beyond merit, but evidence clearly shows the "uncomfortable" result having strong support despite desperate attempts to debunk.
In short; the op's story only works if including the DZA data actually drags the heritability estimate down into trivial territory. It does not.
Later analyses of the MISTRA sample that explicitly include both MZA and DZA twins and use full structural-equation models estimate heritability of general intelligence (g) at about 0.77 in adults. https://laplab.ucsd.edu/articles2/Lee2010.pdf
That is higher, if anything, than the original approximate 0.70. In other words:
The alleged “suppressed control group” does not turn the result into “no heritability”.
The more sophisticated models using that very same DZA data still say “IQ differences in this adult sample are heavily genetic.”
You just don't like this data, and don't want to accept it, because of the implications.