> Sure we can't see dark matter (it doesn't interact electromagnetically), but we can see its effects
Even this is granting too much: "seeing it" and "seeing its effects" are the same thing. No one has ever "directly seen", in the sense that internet DM skepticism demands, anything other than a photon.
The problem with dark matter is that there does not exist any second relationship from which to verify its existence, like in the case of normal matter, which takes part in a variety of interactions that lead to measurable effects, which can be compared.
The amount and the location of dark matter is computed from the gravitational forces that explain the observed movements of the bodies, but there are no additional relationships with any other data, which could corroborate the computed distribution of dark matter. That is what some people mean by "seeing".
All major DM candidates also have multiple interactions: that's the WI in WIMP, for instance. In fact I don't know that anyone is seriously proposing that dark matter is just bare mass with no other properties - aside from the practical problems, that would be a pretty radical departure from the last century of particle physics.
No interactions have been found, despite a lot of resources put into the search. So currently all dark matter particle theories apart from "non-interacting" have been falsified. And non-interacting theories are probably unfalsifiable.
Radical departure may well be needed, for other reasons too.
> The problem with dark matter is that there does not exist any second relationship from which to verify its existence.
This is exactly it! Dark matter is strictly defined by its effects. The only 'theory' part is a belief that it's caused by yet to be found particle that's distributed to fit observations. Take all the gravitational anomalies that we can't explain with ordinary matter, then arbitrarily distribute an imaginary 'particle' that solves them: that's DM.
The problem is that the language used to talk about DM is wrong. It's not that DM doesn't interact with EM, or the presence of DM is causing the galaxies to rotate faster than by observed mass. These are all putting the cart before the horse. What we have is unexplained gravitational effects being attributed to a hypothetical particle. If we discovered a new unexplained gravitational property, we would merely add that to the list of DM's attributes rather than say "oh then it can't be DM".
All physical entities are defined by their effects! Suppose we found axions and they had the right mass to be dark matter. Would that mean we now "really knew" what dark matter was, in your sense? No, it would just push the defining effects further back - because all an axion is is a quantum of the (strong CP-violation term promoted to a field).
Just like the electromagnetic field is the one that acts on charged particles in such and such a way, and a particle is charged if the electromagnetic field acts on it in that way. There's no deeper essence, no intuitive "substance" with some sort of intrinsic nature. All physical properties are relational.
Even this is granting too much: "seeing it" and "seeing its effects" are the same thing. No one has ever "directly seen", in the sense that internet DM skepticism demands, anything other than a photon.