The elite ones will probably be neutral to favorable. Certainly there will be a lot of individual administrators who feel strongly about education and want to make it accessible to more people.
I doubt competition will hurt the private ones that much. They already have orders of magnitude more qualified applicants than they can accept, and are making very difficult decisions to turn a lot of them away whom they'd love to have.
The for profit ones mainly target students who couldn’t get into public universities. The elite ones target students who use public universities as backup. I think some of those students (particularly non wealthy students) will switch to making the elite universities the backup, contingent on the financial aid package. And I think that will have atleast some effect on administrative budgets. The elite universities maybe non profit, but in practice they are run more like hedge funds.
It students are willing to pay $20k more / year for private and public cost goes down by 20k, the private ones will be less competitive unless they also reduce their tuition proportionately. No ?