The study you linked to doesn't find heredity to be the driver of decline of IQ (which they pointedly reminded is not a measure of intelligence):
Regardless, we believe education, test-taking, and media exposure emerge as potential moderators for explaining the observed gains in three-dimensional rotation scores and declines or stagnation in matrix reasoning, letter and number series, verbal reasoning, and composite ability scores.
And did I say or even imply it was? Height is comparably heritable to adult IQ (and coincidentally is even strongly correlated with such) at around 0.8, but obviously somebody who is e.g. malnourished as a child will never reach their height (or intelligence) potential. And that's precisely why heritability is useful. It lets you create a ballpark estimate for the overall average influence of genetic vs contemporary environmental factors. So genetic factors would tend to explain about 80% of the differences in heights, with environmental factors taking up the remaining 20%. The heritability for factors without a genetic component is zero.
Regardless, we believe education, test-taking, and media exposure emerge as potential moderators for explaining the observed gains in three-dimensional rotation scores and declines or stagnation in matrix reasoning, letter and number series, verbal reasoning, and composite ability scores.