Dark matter is invisible, but it isn't magic. It is not significantly different from neutrinos. No one seriously denies the existence of neutrinos nowadays, even though they are invisible (i.e. they don't interact electromagnetically).

Dark matter is actually a very parsimonious theory. None of the laws of physics have to change to accommodate it, unlike with MOND. We may not see it, but it has to move around and affect normal matter in predictable patterns consistent with our current understanding of physics. If it doesn't, then the theory is wrong and may need some revision (which may be a dark matter + MOND hybrid).

In parallel with the research that attempts to find the properties of dark matter that best describe our observations is research that attempt to find what other properties it may have. It is a new particle? Can it interact in ways other than gravity? We didn't find anything, but the universe is under no obligation to make things easy for us.

One possible idea called the "nightmare scenario" is that dark matter is made of particles that only interacts gravitationally. It is a perfectly fine theory, maybe the cleanest one, but unfortunately, it would mean that we may never be able to detect these particles because gravity is so weak that the required detectors would be way beyond our technological abilities.