- Does the experiment prove that the (luminiferous) aether does not exist?
No, it just proved that the luminiferous aether is not needed. The wikipedia article is correct. Physicists were working on the problem of the apparent conflict between Maxwell's electrodynamics and Newtonian mechanics prior to Einstein's development of special relativity. There is an alternative to special relativity, Lorentz ether theory. The latter says that the ether does exist, but that it is not observable. There is no difference between the two theories in terms of predicted experimental outcomes. There is a huge difference between the two in terms of the assumptions those theories make.
Einstein simply took Maxwell's equations at face value: That the speed of light is the same to all inertial observers, with all the repercussions that has with regard to the limitations of Newtonian mechanics. The modern view is of course that the luminiferous aether is not needed, bolstered by the fact that photons do just fine in vacuum
- Why was the aether not observed in the MM experiment?
Because it doesn't exist.
Physicists at that time looked at electromagnetic radiation as being a wave phenomenon, somehow analogous to other wave phenomena known at that time. All wave phenomena known at that time needed some kind of medium to transport the waves. Physicists at that time did not know about quantum mechanics. Photons do just fine in vacuum.