While this issue this is clearly beyond the scope of the 14th ammendment, I suspect most Americans would strongly support legislation to ban the use of family relations in university admission decisions where public funds are in play.
If the Supreme Court interpreted the statutory language banning racial discrimination in federally-fund education the same way it interpreted the same language in employment, it would already be banned as disparate impact discrimination.
Also, I disagree that it is clearly beyond the scope of the 14th Amendment: yes, I’d agree superficially, but then if you took the main opinion in the recent cases at its word when it claps back at the dissent suggestion that universities could move to other bases which are not, but correlate with, race about the ruling banning racially-discriminatory treatment regardless of how it is framed, the current court would absolutely strike down legacies if they were properly before it.
(I predict that if such a challenge comes, in a case where they’d have to hear it to preserve legacies, they will tie themselves in knots to do so rather than act consistently, but that's another issue.)
I barely even need to point out that Republicans didn't need to be scared into opposing legacy admissions, as they eagerly supported them, for people who might be taken in by their meritocratic facade from the decision yesterday.
This is all documented in The Price of Admission (https://www.amazon.com/Price-Admission-Americas-Colleges-Out...).