Maybe we’re referencing different studies. Which are you specifically referencing?
"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."
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.
The "no evidence" comment is not made by the paper itself - it was made about a much weaker 2008 study. The linked paper actually claims that legacy admissions do increase alumni giving by implicitly favoring wealthier families. I think both papers have flaws.