The experiment disproved the existence of stationary aether; it didn't disprove the existence of an aether dragged by the Earth.
Relativity is radical. If we interpret it through the concept and language aether, it says that when two observers traveling at different velocities briefly meet at the same time and place, they will each observe an aether that appears to be dragged with them. That is a contradiction: how can the aether be simultaneously dragged in multiple directions at the same spot? So there cannot be an aether. But there must be something else, all the new cruft contained in Relativity, like dilation of time and space.