An argument that people put forth is that because neutrinos can undergo changes in lepton flavor midflight, they must "experience time." Since massless particles must travel at the speed of light, and since particles traveling at the speed of light cannot "experience time," it follows that neutrinos must have nonzero mass. See this post.
This argument makes sense at first sight, because worldlines of massless particles have zero proper time, given that the worldlines are light-like.
However, based on this post and this post, it is possible to parametrize light-like worldlines in such a way so that $dx^{\mu}/d\lambda = p^{\mu}$ where $p^{\mu}$ is the momentum. Given that both $x^{\mu}$ and $p^{\mu}$ respect Lorentz transformations under changes of inertial reference frames, it follows that the definition of this parametrization $\lambda$ is independent of the inertial frame it was initially defined in.
Now if this kind of parametrization is possible, then I don't see why massless particles traveling at the speed of light can't undergo internal changes. It seems like we can change our notion of time from proper time $\tau$ to this new parametrization $\lambda$.
Experiments have shown that neutrino oscillations definitely occur, but is there a deeper argument that goes beyond special relativity for why that implies they must have mass? I honestly don't see the implication.
PSE pages cited:
