What is the argument for Plessy v. Ferguson (1896) upholding segregation for more than half a century?

it permitted separation of races if facilities were equal

Explanation:

The opposition of the segregation argued that the act of segregation is a direct violation of the constitution. This amendment acknowledged black people/former slaves as citizens.

Which means that they should receive the same right as other citizens. Including public facilities.

According to Judges in Plessy v. Ferguson, The act of segregation is not considered as a violation of the amendment. Because the court at that time believe that the black citizens still treated as 'equal', they're just separated.

This is why the phrase "Separate but equal" become popular after Plessy V. Fergusson.