If they're receiving government money/kickbacks/tax breaks, then to a degree... yes.
For the same reason I don't like the idea of public schools going rogue and not having at least a somewhat centralized basis of accreditation or certain universal standards.
It would seem as if people tend to have a different view on the topic between HS and College.
If the "get in-line with the social policies of this administration, or else risk losing funding" is the kind of threat that people object to, I wonder how they felt about this:
Obama Administration says school districts must not discriminate
time.com
The guidance signed by Justice and Education officials, while not a legal decree, implicitly threatens noncompliant schools with lawsuits or the withdrawal of federal funding.