That is correct as far as the known neutrinos go. If there is a fourth generation of matter, however, all bets are off. (I'm too lazy to look up the limits on that search at the moment.) On the other hand, since neutrinos oscillate and the sun flux is one-third what we expect rather than one-fourth, you need some mechanism to explain why this fourth generation doesn't show up in the oscillations. A large mass is probably helpful for that, though, if I remember correctly.

Compared to axions or supersymmetric particles, or WIMPs, massive neutrinos have have more of the comfort of home.

Point of order! A massive neutrino is a WIMP. "Weakly Interacting" - that's neutrino to you - "Massive Particle".