..I just don't see how any Aether Concept today would help us understand, even the nature of this universe we live in..
The AWT says, the Universe is basically of chaotic nature of many quantities, the pair of them (inertial energy and mass density) is relevant for us due the anthropic principle. You can think about it as about solution of question: how the ideal shape and density of inertial gradients should appear to see as most from our Universe, as possible. This question is quite relevant in dense fog of inertial particles. If such fog will be too dense, you will not see many particles because of dispersion. If the fog will be too sparse, you'll see at the distance, but you'll see a too few particles anyway. So here's some optimum density of fog, which will enable you to see as much, as possible.
..What experimental evidence do you suggest?..
For example, this is the scheme of
MAGIC experiment (dispersion of gamma light), which is relying to the spin foam structure of vacuum. In fact, the same reasoning is working even for inertial foam, too. But the AWT is predicting the same result even for the case of microwaves, which are expected to be dispersed by the vacuum foam as well.
The another question is the intepretation of well known reality. Can you explain, why the light is spreading in transversal waves at the distance? If the vacuum would be of zero density, then the frequency of energy waves would be infinite. Why just the transversal waves? We know, in sparse particle encironment the energy spreads along longitudinal waves. But as the environment density increases, the character of energy waves will change to the transversal one.
When the density of vacuum foam increases (for example near magnetars and black holes), we can observe the gradual switching to the transversal model of energy spreading again, followed by the birefringence of vacuum, multiplication of event horizons and many other related phenomena. The system of testable predictions of AWT are quite broad and complex.