Is this just a historical artifact - that the particle physics community decided at some point to call all of the pre-oscillation physics by the name the "Standard Model"? The reason I ask is because I often see articles and books say something to the effect "the strongest hint of physics beyond the SM are the non-zero neutrino masses" as if this is something significant and mysterious - whereas from what I gathered from the answer to a question I asked previously , lepton mixing is something natural and unsurprising. So why aren't neutrino oscillations considered part of the SM? I am not asking out of any sociological interest but because I want to make sure I haven't underestimated the significance of the discovery of neutrino oscillations.
The historical formulation of the SM involved one Higgs doublet and only renormalizable couplings, the latter being due to the focus at the time on achieving a renormalizable formulation of the weak interactions. With these restrictions neutrinos are massless and do not oscillate. To get neutrino masses you need to extend this framework either by adding non-renormalizable dimension 5 operators, which one would naturally expect to be there in the framework of effective field theory, or you have to add renormalizable couplings involving new fields, typically including SM singlet Weyl fermions (i.e. right-handed neutrinos) and a SM singlet Higgs field. How much of an extension of the SM this really involves is subjective. There were many theoretical papers speculating on such extensions before the actual discovery of neutrino oscillations.
Because neutrinos were still widely considered massless at the time "The Standard Model" was formulated.
One could argue that we're on StandardModel v2.3ish at this point and that the up-to-date release includes massive, mixing neutrinos, but that just leads to a confusion of terminology.
Because there are different extensions you can use to give mass to the neutrinos. You can put mass only in left neutrinos, or you can add right neutrinos, and it is even unclear how many species to add. Of course, a GUT-like neutrino, as in SO(10) etc, seems preferable, but it is not the only option.
I'll add reference to the "Ten Lectures on the ElectroWeak Interactions" by Barbieri. In my opinion -- the best reading on the electroweak physics.