If a private school accepts, like most colleges do, any form of public funding, it can't discriminate.
They can't discriminate based on race, religion, etc. But they can have admission standards like many colleges do. Many charter schools exist to get state money and are not required to meet many state standards. Many go out of business and cannot account for their money (as is true for private for-profit colleges that exist on federal loans and grants). Biden just eliminated billions in debt for students who attended for-profit colleges that went out of business in the middle of the term.