You are thinking that Brown v. Board of Education voided Plessy, but it did not.
It created an EXCEPTION to the general rule of segregation in the case of public schools, based on "recent psychological research" which allegedly showed that applying the perfectly valid and Constitutional rule of "separate but equal" was not permissible IN PUBLIC SCHOOLS because the "research" showed that the "mark of inferiority" associated with separate schools was the cause of poor performance by black students.
But don’t you think that the SCOTUS would strike down any attempt to segregate in public facilities based on the Brown v. Board of Education ruling? If this is so, then how is it really just an exception for public schools.