The rights of "private" clubs, schools, organizations have long been preserved by the courts.
Think about religion as a private club. If you don't hold with the views of the church, they'll throw you out. Any business can refuse to do business with anyone for whatever reason...gyms,...country clubs...consultants...do it every day.
If a "private" school has its own rules, and does NOT receive federal funds, they can pretty much do whatever they want. You can have all white girl schools, all boy schools, and so on. If fact, those very cases exist all over the country.
The US Boys Scouts is an interesting organization. BSA was created by Congressional charter and receives lots of accommodation from a variety of various government agencies, BUT...the BSA forbids homosexuals, and pushes religion. BSA, based in Irving, Texas, right up the road from me, is heavily influenced by The LDS Church. BSA even had a whole show dedicated to it on Penn and Teller's "Bullshit" on HBO. E.g. BSA rents some federal properties for $1.00.
Should an agency that receives federal funds, and favors, be allowed to discriminate? Most of us would say no, I think.
Now, should Black-only colleges be allowed? Should boys only schools be allowed? Should religious schools be allowed to teach their own doctrine as they see to fit and to establish certain customs and conduct? Most of us would probably say yes.
Freedom typically cuts both ways. Private organizations are private for a reason. They are MEANT to be exclusive. As long as they don't take from the public coffers, or get a load of tax breaks, I think they can do what they want within a range of sensibility. Of course, few of us would argue that the KKK is not a nice private group, but...here's the kicker, should we deny them the right to assemble because the majority hold them in disapproval?
Churches are an interesting case. They're tax free, for the most part, yet, support ideas many of us don't approve of. Should that be allowed? Maybe not. What about the ministers making millions? Should that be tax free?
Churches and the BSA hold a fascinating place in the system.
I say, treat everyone the same, with regard to taxes, and tax breaks, and discrimination law. Apply the rules the same across the board. The law of the land is not that way though.