The "private clubs" that were forcibly integrated were done on the grounds that they were "private" in name only. There still are a number of "no blacks" private golf courses in the US. If you claim the "private club" exemption, you really need to closer to "private" than "public".
You are quite right here. I did not think it necessary to make that distinction, as it was not the main point I was arguing. But staying with that point, the forced integration of private clubs was a very short lived controversy as I recall it. The establishment forces quickly backed off on pursuing such cases by government action, as there was a strong backlash to such actions. Civil rights groups then focused on using social pressure, and did so quite successfully for the most part. At least that is how I recall it.