Oops.  Accidentally dropped a paragraph about whether you could increase height/SAT scores across the board with the existing policy levers.  In that case, a high heritability doesn't directly say that that's unlikely, but you would at least have to expect to increase your policy setting to the point where most people are getting more of the nutrition or education or whatever than is currently a few standard deviations above average, since otherwise if less than that much of your policy was having an effect, you would expect to already see it in the heritability measurement.  So heritability doesn't say that feeding kids 10,000 calories a day or sending them to school for 12 hours a day won't have effects on height/SAT scores, but as a general rule you'd be looking at such extreme interventions that it's probably not a good direction to go.

It is not that intuition.  The mainstream intuition is that a high heritability means that variations in the thing that you're measuring (height, SAT scores, infant mortality) is primarily affected by genetics and cannot be influenced by the environment.  A better framing would say that high heritability means that variations in the thing that you're measuring are not well explained by existing variation in the environment.  To reframe, then, what I was going for above, a high heritibility is an upper bound on how much you can expect to improve a specific measure by playing around with existing policy levers, versus having to think outside the box and develop new policy levers.

So, again, pretend that SAT scores have a high measured variability in the US.  (I think it's actually closer to 50 than either extreme, but I don't have off-hand an actual example of a measure that we might want to increase/reduce the variance of as a society that does have a high heritability.)  We might look at nutrition, SES and education and ask whether we can manipulate these to significantly reduce the variance in height and/or SAT scores.  Under the high heritability assumption, the answer is that this is very unlikely for both stats.

Again, this is without saying that high heritability means the existing policy levers are not having an effect right now.  If we, as a country, stoppd feeding kids at age 5 and left them to fend for themselves, I would expect the survivors to be both shorter and have lower SAT scores than preceding generations.  If we stopped sending kids to school, I would expect average SAT scores to drop and the variance to increase significantly.  And nothing about heritability argues against any of that.

It also doesn't mean that there are no policy levers that could have an effect.  I am not biologist enough to know if the application of HGH to children on a large scale would increase average height, but what I can say is that heritability has nothing to say on the question, since it is currently an environmental variable with almost no variance itself.