Why are extremely small masses expected to be in relativistic motion?

The most common neutrino, here, now, is the solar neutrino. The spectrum is:

so mostly around 300-400 KeV. We don't know the electron neutrino mass, but let's say it's around $m_{nu}=0.07,$eV, that puts the Lorentz factor (in the Sun's rest frame, which we are approximately in):

$$ gamma_{nu} =frac E {m_{nu}} approx 5,000,000$$

which is squarely in the "ultra-relativistic" category.

In short: the nuclear transitions that produce neutrinos have energies that are much, much, greater than the neutrino mass.

Also: since they are weakly interacting, they don't thermalize with their surroundings, expect perhaps in extreme cases (high density) like core collapse supernovae and the Big Bang.