"... and whether the institution or facility prohibits persons from entering, and fully participating in an integrated manner"

exactly. No school that I know of is prohibiting persons of any color from entering. THAT would be illegal.

....well....except that, when the government interferes by forcing integration, THEN people ARE prohibited from entering the school of their choice. That is exactly why the supreme court ruled against the forced social engineering in Seattle. It was determined that, on the basis of race alone children were forced into a school not of their choice -- or said as you worded it -- children were prohibited entry to the school of their choice on the basis of race alone.