Neutrinos with mass certainly DO fit in the Standard Model. In fact, all 3 known left-handed neutrinos are a part of the standard model. Neutrinos are even known to oscillate between the 3 types. Originally, neutrinos were assumed to be massless as their mass is so incredibly tiny it couldn't be detected when the particles were first proposed and discovered. Their insignificant mass didn't alter any predictions the model made on particle physics at the time. That does not mean that they aren't more well understood today, nor that they have some magical capability that doesn't fit the framework of the standard model. They were just assumed to be massless because they moved at near light speed and there didn't seem to be any right-handed neutrinos detected that would show they interacted with the Higgs... also, every particle interaction that created them didn't have any missing mass that would need to be accounted for by the neutrino ejected from the collision.
There are theories on how and why the neutrinos oscillate between the 3 mass states and on how they interact with the Higgs to generate those masses. There are even theories that include right-handed "sterile" neutrinos that we haven't yet detected (and possibly can't ever hope to detect based on theories of their properties.) The fact that we can't prove it and don't have any good experiments in progress to figure it out doesn't mean the 3 flavors of neutrinos with their various masses don't fit perfectly well into the standard model as-is. These tiny, fast, ghostly particles just don't interact with regular matter very often, nor do they interact with electro-magnetic fields... so, it's very difficult if not impossible to devise experiments to definitively tell us much about how they generate their masses from the Higgs (or some unknown source) or why there are no detectable right-handed neutrinos (assuming they even exist... and if they do, that they exist for long enough to be detected before flipping back to left-handed ones).