Although heritable diversity is bad from an egalitarian perspective, the goal is presumably levelling of outcomes, not of heritability in itself. Turkheimer says bad environment reduces heritability; makes sense. Then improving worst environments both increases heritability and reduces outcome variation. This is an improvement of the worst shared environment, not the worst self-selected. Is that different? The shared environment is what children in the same family have in common. This can't include self selected unless everyone is the same in their preferences. Similar issues probably arise with GxE interaction in the twin study. But to say that we “improve” the environments available for self selection implies a ranking of environments that may be less plausible for free choices.
There's also the argument that people may need to be forced or nudged into investing in education because it's “for their own good”. Many people may be motivationally poorly suited to the modern world, especially children. This is the classic coercive aspect of education. That is, social control over self selection is already accepted to some degree. This reduces the contribution from innate motivation, so would reduce heritability of educational level. Can you really make the horse drink? Not deeply, but the reduction in SES heritability is real, as is some reduction in outcome variation, just because the rising mean pushes the distribution up against a wall. Motivation + ability has a bigger effect than just ability alone.