They intentionally missed her point so they could mock her, following a lead set by her opponent. People are accusing her of stupidity based on an argument she didn't make, so maybe HER high ground isn't in question here. Separation is well established, but not in the Constitution, and the whole principle behind the Constitution is that any legal principle, however venerable, is illegal if it conflicts with that bedrock document, because the Constitution reserves anything further to the states and the people. That's why the SCOTUS heard Plessy v. Fergusons challenge to segregation, and also why a different SCOTUS overturned that ruling 80 years later. Both times the argument hinged on the Equal Protection Clause of the Fourteenth Amendment, but if even SCOTUS rulings weren't subject to Constitutional tests Jim Crow would still be the law in a third of the country.

The Constitution is the highest law of the land; additional laws and court rulings are valid insofar as they're consistent with it, but always subject to review and reversal if found not to be. Ironically, it's the same argument many conservatives dismiss as "judicial activism" but it's nearly as old as the country (though, oddly, judicial review is defended as an "implied" power asserted by the very first SCOTUS but never explicitly stated in the Constitution).