Or, you know, as aether.
It's a scientific theory. It's the best that we have right now to model the real world and be able to do prediction on its behavior.
Does it seems to be kept together by duct tape? Maybe.
Is it yet useful? Yep.
Will it be discarded if anything more fitting will came up? You can be sure of it.
Yeah, pretty much, which is why this adherence to dark matter seems even more puzzling: we already had a mysterious substance with nonsensical mechanical properties (perfectly solid, but has zero collision) that turned out to be completely superfluous; the actual answer was the different shape of the physical laws. Now we again have a mysterious substance with nonsensical properties (has gravitational pull, doesn't interact with normal matter in any other way) — could it be that it simply doesn't exist?
And it's not like the concept of aether itself was really all that useful for anything. The physicists wanted the light to have some mechanical medium to propagate through instead of being a thing of itself, that actual itself shaped mechanical media, not the other way around (mechanical properties arise from the E-M interaction, not the other way around), simply because all other known waves phenomena existed in mechanical media.
Of course, it could be that dark matter does not exist. In a very real way, nothing in physics “exists” because like all natural sciences physics does not make statements of objective truth, it makes testable predictions.
Dark matter, string theory, aether, etc., those are models that we, at some point in time, think may help us get better predictions and design further experiments. All models turn out to be wrong in the end, but they can be helpful until we come up with better ones.
If you drop the dark matter model, then you would want to have some other model as for why we observe what we observe. Some people find that other available models are even worse than the dark matter one, but if you don’t think so you can take your pick.
Except that there is nothing nonsensical about a particle that has mass and doesn't participate in any SM interaction. It's inconvenient if such a particle exists, as it's very very hard to detect things precisely by their gravitational effects, but there is nothing nonsensical, or even particularly weird, about the idea. Plenty of particles only interact with a few of the SM forces - e.g. photons are not affected by the strong force, nor are electrons, neutrinos are affected by neither the strong force nor EM, only the weak force, gluons only interact with the strong force, not EM nor the weak force, etc