The question you ask—are these opportunistic moves zero-sum?—is still an interesting one. But the realities of the sub-population studied here change the intuitive answer to that question. There may be a finite number of seats at Harvard, but there is plenty of capacity for upward mobility from lower classes into the middle class.
[0] p864 here for income statistics https://scholar.harvard.edu/files/lkatz/files/chk_aer_mto_04...
On the other hand, it doesn't have to be a requirement to happen. Making some people smarter can actually have a negative effect on the smarter ones.
This can be illustrated with schools, where it benefits the troublemakers to be placed in a group with calm children, but it is on the expense of the calm children who will now be disrupted in their studying. In the scandinavian countries, this is often done to children. Troublemakers, most often boys, are placed together with groups of calm children, often girls. This is such a common practice that there is an expression for such girls - pillow girls, "kuddflickor".
https://translate.google.se/translate?hl=sv&sl=auto&tl=en&u=...
That said, isn't the Chetty proposal (moving from low-opportunity location to higer-opportunity location) basically why there are millions of migrants in Europe right now.
Girls are generally expected to be better behaved. This can be extremely suffocating and unhealthy.
If they are exposed to "troublemaker" boys and learn to have a little back bone and stand up for themselves (to possibly oppressive adults), it's still less socially acceptable for a girl than for a boy. So then she blames the "troublemaker" boy to cover for herself because he will get in less trouble than her and that's what the adults find plausible anyway.
It's entirely possible that even if she tried to accept the blame, the adults would interpret that as her trying to be kind to him and cover for him.
If you remove all the smart people from a school I'm pretty sure the outcomes for everybody left behind will be worse.
As the parent and other sibling comments point out, the real advantage may come from positive-sum effects: reduxed inequality, a more efficient market from a leveled playing field, etc. But that doesn't change the fact that the math being used is too incomplete to be held up as an economic efficiency argument, the way the article does.
This logic assumes that none of this benefit is positional. The family moves to a richer area, the kids do well in the fancy local schools, get into top colleges, make lots of money, etc, yes. But a lot of these opportunities look like "each year we take the best N people we can", so there are probably other (less qualified) kids who do slightly worse than they would without the move.
As an analogy, if being in the tallest 5% of society makes you earn $100k more, we would expect a randomized controlled trial on increasing children's heights to have a large positive effect on income. But if we then rolled that out to the whole society those benefits would turn out to be illusory.
This is still valuable from a perspective of reducing segregation and inequality, but I'm skeptical of its presentation as an extremely good deal economically.