Surely the idea of it being a new kind of matter that interacts gravitationally but not electromagnetically yields some testable result? Does it actually yield nothing testable with today’s experimental methods?

I'm not sure it's inherently unfalsifiable. There are some specific proposals for dark matter that could be ruled out by experiments, such as right-handed neutrinos: https://en.wikipedia.org/wiki/Sterile_neutrino#Sterile_neutr...

Maybe if you're being very broad in definitions then some class of proposals describable as "dark matter" might be unfalsifiable, but to be taken seriously as a scientific proposal I think it should be specific, concrete, and indeed testable, and there are a few of these within the "dark matter" class.

Again, we're in the perhaps unsatisfying position of having observations which don't cohere with our current theoretical understanding. What's the solution? It's not easy...

Is the existence of a planet so easily falsifiable? It hasn't been so long since the Planet Nine hypothesis started going around, and while we've observationally ruled out a big chunk of the original parameter space, there's still lots of room for a big dark dwarf planet to be floating around out there. It doesn't seem so different from how we've gradually been ruling out the parameter space for dark-matter observations.

I mean, dark matter may be discoverable, we just don't know how if it exists. There was time between the irregularities that were noticed in the orbit and the discovery of a new planet.