I could equally (and accurately) argue that in practice public school means discriminating against people by wealth, using zip code as a proxy.

And anyway, why would "discriminating based on competence" exacerbate inequality? Shouldn't competence-based discrimination lessen inequality e.g. by giving bright, poorer schools a shot at more prestigious institutions while financially-but-not-intellectually privileged kids move to more middling institutions?

Given the choice between wealth discrimination and merit discrimination, I would prefer the latter.