> Isn't that the basic objection to dark matter, that there's nothing in it that's falsifiable?

I don't think that's case, at least to me it seems that the basic objection is some vague discomfort with the idea, not an issue with the falsifiability. And dark matter is clearly falsifiable, it has already survived many observations.

The CMB power spectrum would look entirely different if there was no dark matter.

Gravitational lensing in certain areas in the sky wouldn't look as strong as it does without dark matter either.

Until Michaelson-Morley’s landmark experiment, we hadn’t found a way to falsify aether either. Is there physical proof that dark matter is not falsifiable? If not, we still have a bit further to go.

There is always a tension in Physics between physical proof and observability on the one hand, and having the right idea, the right intuition on the other.

Going back to my Einstein example, there were other Physicists who contributed to Relativity (Lorentz, Minkowski, Poincaré, Fitzgerald) but we give Einstein the most credit because he had the right idea, that it's not enough to treat the speed of light as a constant in your math, you actually have to believe the Universe works that way. One of the stories that gets told is the reason Einstein never got a Nobel prize for Relativity is because the Nobel committee just wasn't ready to give it up and credit Einstein with discovering a new, fundamental truth about the Universe.

Another story that gets told is that when Einstein was an undergrad at the ETH in Zurich, Minkowski was one of his professors and he dismissed Einstein for being a "Lazy Dog" because the math never came easy to him. But he looked at how Physics was taught at the time and knew something had to change. And he was right about that.

I'll just leave this here, Feynman explained it better than I ever will.

https://youtu.be/NM-zWTU7X-k

Dark Matter in totality is hard to disprove (as it's really not one theory, but a rather large set). But large parts of that parameter space is testable, and is tested. See WIMPs for example, where we searched extensively. Have we ruled out everything? No. But we have ruled out many variants, mass and coupling ranges.

Michaelson-Morley didn't falsify aether. At the time, their results were interpreted within the aether framework in terms of compression of the aether yielding the Lorentz transformations.

We still have the aether, we just call it the "electromagnetic field"

We absolutely do not have an aether! Light does not travel through a medium like ripples in water.

The aether makes specific predictions. Like the fact that we're all traveling through it because it is a background medium. These were clearly refuted.

Fields are not aethers. Fields do not have a reference frame.

Why wouldn't dark matter be falsifiable?

I mean, i imagine its difficult because we can't really conduct direct experiments, but i don't see any reason that its non-falsifiable.