The problem is that Maxwell's equations predict the speed of any EM wave to be $c$. But this doesn't make sense in Newtonian physics because velocity addition is linear. The aether offered a possible solution because it picked out a preferred reference frame, but if you do away with the aether you're back at the original problem: with reference to what do EM waves have a speed of $c$?
The solution is of course that we need to use special relativity, and in SR velocity addition is not linear.
There is an alternate formulation to special relativity, Lorentz Ether Theory. This alternate theory allows the ether frame to still exist. Nobody teaches it. Why?
Special relativity makes two very simple assumptions, that the laws of physics are the same in all inertial frames, and that the speed of light is the same to all inertial observers. The Lorentz transformation and everything implied by it follow from these simple assumptions.
Lorentz Ether Theory on the other hand posits a special frame, the ether frame, where Maxwell's laws truly do hold. Per this theory, this is the only frame in which the one-way speed of light is Maxwell's c. Lorentz Ether Theory also posits time dilation and length contraction as axiomatic. These lead to the Lorentz transformation, and to the round trip speed of light being Maxwell's c.
The only way to distinguish these two theories is to find a way to measure the one-way speed of light. That's not possible, and thus there is no way to experimentally distinguish the two theories. Yet physics instructors only teach special relativity. You have to dig deep, very deep, to find proponents of Lorentz Ether Theory.
One reason is that the assumptions of time dilation and length contraction as axiomatic seem rather ad hoc (and that's putting it nicely). An even bigger reason is that Lorentz Ether Theory introduces a key untestable hypothesis, the existence of the ether frame. This frame cannot be detected. Time dilation and length contraction conspire to hide it from view. A bigger reason yet is general relativity. The axioms of Lorentz Ether Theory are inconsistent with general relativity. The final nail in the coffin is quantum mechanics, which eliminates the need for a medium through which light propagates. Without that very self-contradictory medium (the luminiferous aether), what's the point of having an ether frame?
The modern geometrical perspective of special relativity isn't so much that the speed of light is constant but rather that there exists some finite speed that is the same to all observers, and that light necessarily moves at this speed because it is carried by massless particles.
What motivates the existence of this finite universally agreed upon speed is geometry. What geometries yield a universe in which Newtonian mechanics appears to hold in the limit of zero velocity, and what do these geometries say about a speed that is the same to all observers?
The answer is that there are two cases: This universally agreed upon speed is infinite or finite. An infinite universally agreed upon speed results in Newton's universe. A finite speed results in Minkowski space-time describing the geometry of special relativity. Experimentally, the finite speed of light appears to be the same to all observers, thus falsifying the notion of a Newtonian universe with Euclidean space and time as the independent variable.
Best Answer
There is none. The aether model is self-consistent. But it violates Einstein's principle of relativity (which states that there is no special frame of reference) which wouldn't be such a big deal, if not for the contradiction with experimental results, e.g. the Michelson-Morley experiment