"Dark matter" is just a description that refers to matter which interacts gravitationally but not, or at least very weakly, in other ways. It doesn't imply anything about the nature of such matter. We
know that there are several types of dark matter, specifically neutrinos and brown dwarfs, although it is now established that they don't make up anything near the density required by the cosmological models. Primordial black holes, which may exist, would be a dark matter. Some particle physics models, like supersymmetry, also naturally predict that there will be a massive particle that would behave like dark matter. So I don't think the analogy with the aether is very good, because that really was pulled out of thin air (pun intended). Given how successful general relativity is, it's perfectly rational to interpret the observations as probing mass distributions while assuming GR will continue to hold. There are of course people also assuming that GR is wrong and attempting to explain the observations without dark matter, but my impression is that they struggle to come up with self-consistent theories that fit all the data, although I'm not very familiar with that work.
What other people have been doing over the past 40 years is attempting to devise tests for these various dark matter candidates. We know, for example, through lensing observations, that MACHOs/brown dwarfs don't exist in the required numbers and that the neutrino mass seems too low. The problem, of course, is that there are only so many ways to try to observe matter that is truly dark.
I agree, though, that, in the end, it may be that dark matter will be an untestable hypothesis, just like quantum gravity or whatever.