This explanation is insufficient. If neutrinos were indeed massive particles we'd see a wide distribution of their velocities, just like we can observe slow and fast protons, slow and fast electrons, slow and fast everything that moves slower than c. Yet, in 100% of experiments that have been done all neutrinos are propagating through space at the speed close to or exactly equal to c.
The reason for this is the extremely small mass of neutrinoes. The current experimentally-derived upper bounds on their mass is around 1 eV (in contrast, an electron has a mass of 511,000 eV). This means that any process that creates a neutrino will give it enough energy to send it off at ultrarelativistic speeds. Even something simple like neutron decay can impart 1 MeV of kinetic energy to a neutrino, which, as the grandparent calculates, means the neutrino is traveling at 0.999999999999*c. Only chemical reactions would release a small enough amount of energy to have non-relatvistic neutrinos. But, chemical reactions don't release neutrinos.
This is why we only see speed-of-light neutrinos. This is also why it's taken so long to discover that they have mass.