Strictly speaking, it is indeed incorrect that neutrinos travel at "close to the speed of light". As you said, since they have mass they can be treated just like any other massive object, like billiard balls. And as such they are only traveling at nearly the speed of light relative to something. Relative to another co-moving neutrino it would be at rest.
However, the statement is still true for almost all practical purposes. And it doesn't even matter in which reference frame you look at a neutrino. The reason is that a non-relativistic neutrino doesn't interact with anything. Or in other words: all the neutrinos you can detect necessarily have to have relativistic speeds.
Let me elaborate. Since neutrinos only interact weakly they are already extremely hard to detect, even if they have high energies (> GeV). If you go to ever lower energies the interaction cross-section also decreases more and more. But there is another important point. Most neutrino interaction processes have an energy threshold to occur. For example, the inverse beta decay
$$ \bar\nu_e + p^+ \rightarrow n + e^+$$
in which an antineutrino converts a proton into a neutron and a positron, and which is often used as a detection process for neutrinos, has a threshold of 1.8 MeV antineutrino energy. The neutron and the positron are more massive than the antineutrino and the proton, so the antinneutrino must have enough energy to produce the excess mass of the final state (1.8 MeV). Below that energy the (anti)neutrino cannot undergo this reaction any more.
A reaction with a particularly low threshold is the elastic scattering off an electron in an atom. This only requires a threshold energy of the order of eV (which is needed to put the electron into a higher atomic energy level). But a neutrino with eV energies would still be relativistic!
Assuming that a neutrino has a mass of around 0.1 eV, this would still mean a gamma factor of $\gamma\approx 10$. For a neutrino to be non-relativistic it would have to have a kinetic energy in the milli-eV range and below. This is the expected energy range of Cosmic Background Neutrinos, relics from the earliest times of the universe. They are so to say the neutrino version of the Cosmic Microwave Background. So not only do non-relativistic neutrinos exist (according to mainstream cosmological models), they are also all around us. In fact, their density at Earth is $\approx$50 times larger than neutrinos from the Sun!
There is a big debate if they can ever be detected experimentally. There are a few suggestions (and even one prototype experiment), but there are differing opinions about the practical feasibility of such attempts. The only process left for neutrinos at such small energies is neutrino-induced decay of unstable nuclei. If you have an already radioactive isotope, it's like the neutrino would give it a little "push over the edge". The $\beta$-electron released in the induced decay would then receive a slightly larger energy than the Q-value of the spontaneous decay and the experimental signature would be a tiny peak to the right of the normal $\beta$-spectrum. This will still be an extremely rare process and the big problem is to build an apparatus with a good enough energy resolution so that the peak can be distinguished from the spectrum of normal spontaneous nuclear decay (amidst all the background).
The Katrin experiment is trying to measure the endpoint of $\beta$-spectrum of Tritium in order to determine the neutrino mass. But under very favorable circumstances they even have some chance to detect such a signature of cosmic background neutrinos.
TL;DR: In fact there are non-relativistic neutrinos all over the place, but they they interact so tremendously little that they seem to not exist at all.