Pretty simple solution: Have these private institutions get together for the purpose of forming their own accreditation standards that are much stricter than those of the US government.
Businesses know a higher standard when they see it, and as is the case with Home Schooled people, will tend to hire these graduates first and look for graduates of schools with the higher accreditation standard. The Government standard will become irrelevant.
Hard to argue the success of a place like Grove City College compared to the government funded institutions of “higher” learning.
I’d imagine the barriers to entry would be very high.