The missing assumptions are that the number of genes is large, independently distributed (i.e. no correlations among different genes), and identically distributed. And the whopper: that nurture has no impact.

You can weaken some of those assumptions, but there are strong correlations amongst various genes, and between genes and nurture. And, one "nurture" variable is overwhelmingly correlated to many others: wealth.

Unpacking wealth a little, for the sake of a counterexample: one can consider it to be the sum of a huge number of random variables. If the central limit theorem applied to any sum of random variables, it should be Gaussian, right? Nope, it's much closer to a Pareto distribution.

In summary: the conclusion of the central limit theorem is very appealing to apply everywhere. But like any theorem, you need to pay close attention to the preconditions before you make that leap.

"Number of genes is large" is what I said, that's not a missing assumption, I said that explicitly.

The nurture/nature relationship to IQ has been well-studied for many decades. There are easy and obvious ways to figure this out by looking at identical twins raised in different homes, adopted children and how much they resemble their birth parents vs adopted parents, etc. Idealists always like to drag out nurture effects on IQ like it's some kind of mystery when it's a well-studied and well-solved empirical question.

It easily includes nature impact for the same reasons: an incredible amount of nuture items are both Gaussian distributed and the population sampled is large.

Wealth being distributed as Pareto would imply its effects on nuture are not Pareto since the effects of wealth are not proportional to wealth. At best there’s diminishing returns. Having 100x the wealth won’t give 100x intelligence, 100x the lifespan, etc. And once you realize this, it’s not far till the math yields another Gaussian.