> can't be directly observed

Cold (as in moving slowly compared to the speed of light) dark matter has yet to be directly observed. We have certainly observed relativistic dark matter in the form of neutrinos, including streams of them we can produce in labs like Super-Kamiokande [1], where we can watch the flow of momentum exceed that available from electromagnetic interactions. "Relativistic" here means moving as speeds closely approaching the speed of light.

It took almost a decade to go from experimentally demonstrating a momentum-deficit in certain nuclear interactions, to proposing a self-consistent and consistent-with-observations theory of an electron-neutrino carrying the "missing" or "surprise extra" momentum in examining clouds of nuclear decay products. There were many theoretical false-starts, where the behaviours of the momentum deficit in the bulk did not match up with microscopic details of the theory. This is in spite of direct in-laboratory repeatability at accessible energies with then-modern equipment. Worse, it took a further twenty years to directly observe the anti-electron-neutrino (consequently also proving the existence of the electron-neutrino) in a controlled experiment.

Although neutrinos individually carry low momentum, and individually interact only rarely with atomic nuclei, there are an awful lot of them carrying momentum out of stars, and even more out of supernovae and other violent phenomena, so a bit more than sixty years after the first detection of a neutrino, laboratories are pretty good at detecting them. Theory has had to adapt to laboratory discoveries, adding in two other types of neutrinos, and neutrino oscillation. Along the way, the theory has become relativistic (allowing for accurate prediction of observations by differently-moving observers) and generally covariant (allowing for accurate prediction of observations by differently-accelerated observers or equivalently observers near different sources gravitation, including neutrino interactions in stellar cores or near the surfaces of neutron stars).

Neutrinos enter into the energy-density of the universe in the standard model of cosmology; they are there and they make a difference to the arrangement of visible matter in the sky.

In the 1880s (yes, nineteenth century!) a momentum-deficit was detected at the scales of clusters of stars in our own galaxy, with a radial relation being proven for the Andromeda galaxy in the 1930s, which was about the same time that the first self-consistent theory for the (neutrino) momentum-deficit for beta decays was produced. Sky-observing technology development did not quite keep up with particle-observing technology over the intervening decades, so it was not until the 1970s that a momentum-deficit in galaxies is extremely commonplace.

There is a self-consistent relativistic, generally covariant theory for this momentum-deficit in the bulk, but the microscopic details are not yet known from experiment. That is, the equivalent of the electron-neutrino has not been shown, and there may be more than one microscopic component, roughly similarly to how there are three types of neutrinos (and three anti-neutrinos), and of course there may be dynamics (similar to neutrino oscillation).

That's why there are so many detection experiments. There are lots of possible microscopic contributions to dark matter in the bulk, but they are all even more difficult to detect in principle than the neutrino, because (being cold in the sense above) interactions will nudge atomic nuclei with much less momentum than neutrino interactions with the same atomic nuclei. Also, we don't have a way to build a source like Super-Kamiokande that lets us study a stream of excess non-neutrino, non-electromagnetic momentum.

However, there is no serious doubt as to the momentum-deficit at scales of galaxies and galaxy clusters. It's not just galaxy rotation curves. There is also a momentum-deficit clearly visible in the cosmic microwave background, and it is an extremely close match to that seen in the more recent universe at galaxy cluster scales. The only non-self-contradictory theories to date suggest an entry into the Einstein Field Equations in the stress-energy tensor. The entry need not be much like a neutrino or any other particle of the Standard Model of Particle Physics, and in principle might not even interact with them except gravitationally. Those possibilities will necessarily be studied later, if and as technology develops.

> I'd really appreciate some clarity on this

The mathematics of dark matter in the bulk has generally followed from observations of mass-deficit. However, the mathematics of various proposals for the microscopic details of the composition of that bulk has outpaced observation and experiment. A number of proposals were motivated by other issues essentially unrelated to gravitation, so the mathematical structure was there in a modification of the Standard Model of Particle Physics, and experiments would simultaneously test such extensions and the possibility that bulk dark matter is (at least partially) such particles.

Thus, on the one hand, annoying observations that won't go away led to "hot" dark matter (neutrinos) ultimately, there are annoying observations that won't go away that are consistent with "cold" dark mater (details not yet known). Known types of neutrinos and their dynamics proliferated once they were directly detectable in a laboratory setting, leading to updated theories. However, there is as yet no direct detection of a "cold" dark matter candidate, and no especially strong reason to believe that an initial laboratory observation of one such candidate will be all there is to discover in the dark matter sector.

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[1] http://t2k-experiment.org/neutrinos/a-brief-history/