I will start by saying that I know very little about theoretical physics so I could be totally off here. But to me the idea of dark matter feels a little like the theory of ether[1]. Basically we observed a behavior we couldn’t explain, so we create a new invisible substance that permeates the cosmos to fill the hole in our scientific world view. It turned out that we just didn’t understand the science well enough, and there was no ether out there in the first place. Dark matter feels like too convenient of an explanation given that we have never detected it.
It's exactly like the theory of the ether. It was the fleshing-out of that theory that made physicists doubt its veracity - a substance must permeate all space, be more rigid than steel yet not impede planets orbiting the sun? Really? The Michelson-Morley experiment put the final nail in ether's coffin - but there were already a number of nails. On the upside, it led to great physics! Let's hope we'll be able to say the same for Dark Matter.
Ether was introduced because, previously, all other wave motion was waving something. It was standard issue. Waves in the ocean? Must be waving something. Waves in air? Must be waving something. Positing that there was nothing there "waving" would be the new thing.
Second, introduction of a "thing," only to find out that it is real is quite common. Witness the neutrino -- originally just a placeholder to make the math work.
How is that any different? Sounds like "dark matter was introduced because, previously, all other matter interaction was pulling on something". It walks invisibly but quacks like matter (lensing, galaxy geometry, etc) so we call it dark matter.
It's all the ways in which it doesn't act like matter which make it interesting.
[1] https://en.m.wikipedia.org/wiki/Aether_theories