No, we're not. It controls for wealth, and then says that the policy itself affects the wealth of alumni, which makes it entirely inappropriate to remove that as a variable. In essence they are arguing that legacy preference systems are not what motivates higher donations, but they are a causal factor in the outcome of higher donations.

"the policy allows elite schools to over-select from their own wealthy alumni. In other words, the preference policy effectively allows elite schools essentially to discriminate based on socioeconomic status by accepting their own wealthy alumni families rather than basing admissions on merit alone."

" we show that prior to controlling for wealth, there is a strong correlation between alumni giving and legacy preferences. This suggests that greater alumni giving at elite schools with legacy preferences is driven by the school’s ability to over-select from their own wealthy alumni populations—not a result of the preference policies themselves inducing additional giving."

I think you are pointing out a related but slightly different problem. The question being asked is not "Does admitting wealthy students lead to higher alumni donations?" but rather "Does admitting legacy students lead to higher alumni donations?".

The latter one is what is being measured when they control for wealth and conclude that the practice of admitting based on legacy status does not result, in itself, with higher alumni donations. But like you said, when it becomes a means to admitting wealthier students, it can result in the former. But the two questions shouldn't be conflated and measuring them as distinct questions is why it's important to control for wealth in the methodology.

They cannot be measured distinctly when one is a proxy for the other.

It seems like they are correlated, but I don't know that one is necessarily a good proxy for the other.

Would you apply that to, say, wealth and SAT scores?