Issues:
1) In the attached image you have two distinct beam paths: l1 and l2. If you have variable aether modifier k you end up with the travel times t = (l*k)+(l*(1/k)). In each case the scalar k cancels itself out precisely by having the beam inverted for each moment of time. This is to say: whether there is aether motion relative to an observer or not, this experiment CANNOT possibly detect it.
2) The issue is that it presume relative aether motion from the viewpoint of an observer. This, while not necessarily wrong, is fucking dumb to assume for two reasons.
2.a) We're traveling at about 30km/s through the solar system, which is moving about 230km/s through the galaxy, which is moving at about 600km/s through the universe. This adds up to an expected ~860km/s we're moving through the universe, at variable rates in different directions. If there were ANY kind of aether drag (i.e. anything capable of influencing matter) it would have to be undetectably small (even by the difference in momentum light would impart on surrounding matter.)
2.b) If there were any detectable aether drag we've been flying through the universe for some 13.8 billion years. The photon has an incredibly small amount of inertia it imparts when emitted or absorbed, but it is nowhere near small enough that we would not have come to a stop by now.
3) If we're floating in an aether which carries with it inertia in any degree (i.e. capable of AT LEAST impacting the speed of light to a detectable level) then we would have developed a pressure bubble of aether by this point (remember the whole idea is based on waves traveling in water - waves by nature have pressure associated and there is no reason to suspect that giant waves don't exist just as tiny photon-sized waves would.) There would be no relative motion on Earth, or even in the solar system at a detectable level because we would be effectively sitting inside an aether pressure bubble of uniform motion.