One particular luminiferous (just meaning "light-bearing") aether interpretation of the constant speed of light, the Lorentz aether (or ether) theory (LET), is not wrong. Other theories about a luminiferous aether are wrong, since they always lead to the conclusion that the speed of light is not constant in every frame.
The LET is experimentally indistinguishable from special relativity1, and hence as "right" as special relativity. But there are two main reasons to prefer SR as the theory we work with:
The assumption of an undectable "preferred reference frame" by LET is unnecessary - we can just start with assuming the principle of relativity, all this assumption does is add another unphysical, unmeasurable axiom to the theory. By Occam's razor, we thus prefer SR over LET
SR, when formulated in terms of the Minkowski metric, generalizes nicely to GR, almost naturally. There is no obvious path from LET to GR.
It is important to stress that, if two theories have the same experimental predictions, then neither is wrong. Yet, "undetectable aether" is experimentally indistinguishable from SR's perspective of "no aether at all" and to assume the "existence" of undetectable objects when you've got a theory that doesn't need their existence is just ill-advised.
1Note that it is special relativity that predicts a "constant speed of light" or rather, "upper bound of speed of frames into which one may Lorentz transform". Whenever you read "Lorentz invariance", it is about special, not general relativity. In general relativity, all coordinate transformations are allowed, and the "constant speed of light" is recovered by noting that GR locally is SR.