The Michelson Morley experiment was in the 1880s and Tesla held on to the "luminiferous aether" theory until he died in the 1940's I believe. I seem to remember that he thought all other theories were "flawed" like you said.
People will say "Tesla was smart but not a good businessman, etc" and obviously he was a smart guy but still a tragic figure for holding onto old theories some of which bankrupted him.
Belief in a luminiferous ether (that fills all of space) is consistent with the idea that EM waves can travel through space (after all, we can see light coming from the Sun, etc.) and that radio waves could be used.
But Tesla seems to have argued that working "radio" technology was really electrical currents traveling through the ground (or through the ocean in the case of radio on ships), which then somehow induced "reception" in the radio equipment located above the currents.