Harvard also doesn’t want to become a school exclusively for the children of successful Chinese immigrants which is what it would be if it went strictly by test scores and GPA. And for what it’s worth, those Chinese parents don’t want it to be either. As others have noted one of if not the main advantage of attending an ivy is networking with the kind of people whose families can ensure them a spot at Harvard and beyond.
Which one is it? Maybe it's not so heritable.
One possible source of confusion is mistaking “heritable” for genetic. But a heritable trait can be cultural too. For example a child with Spanish speaking parents is considerably more likely to have the speaks Spanish trait. Similarly, Asian immigrants to the USA have an admirable cultural bias toward studying hard and their average test scores have skyrocketed over the past couple decades, far too quickly for it to be a genetic effect.
Now you're lumping Asians, who the admissions system has systematically and openly disadvantaged, with legacy admits by both calling their ability to attend a school as a heritable trait. Yet in one case it's actively discriminated against, and the other it's given preferential treatment. If the goal were to control against the 'heritability' of educational obtainment, wouldn't the primary factor in admissions be weighting parental educational obtainment negatively?