All of physics is a "math model". One we attempt to falsify. And when a scientist does prove some part of the model wrong, the community leaps up in celebration and gets to working on the fix or the next.
Dark matter started as exactly a catchall designed to make the model work properly. We started with a very good model, but when observing extreme phenomenon (in this case the orbits of stars of entire galaxies), the model didn't fit. So either there was something we couldn't see to explain the difference ("dark" matter), or else the model was wrong and needed modification.
There's also multiple competing theories for what that dark matter is, exactly. Everything from countless tiny primordial black holes to bizarre, lightyear-sized standing waves in a quantum field. But the best-fitting theories that make the most sense and contradict the fewest observations & models seem to prefer there be some kind of actual particle that interacts just fine with gravity, but very poorly or not at all with electromagnetism. And since we rely on electromagnetism for nearly all of our particle physics experiments that makes whatever this particle is VERY elusive.
Worth observing that once, a huge amount of energy produced by stars was an example of a dark energy. Until we figured out how to detect neutrinos. Then it wasn't dark anymore.
In short, you're exactly right. It's a catch-all to make the math model work properly. And that's not actually a problem.