I think you're correct to say a lot of people simplify what the problem is with neutrino mass. In principle it seems like there is no problem, you just add a mass term for the neutrino just like any other particle. Just b/c at first we didn't expect that term to be there doesn't mean it's a problem to now, or that original expectation was all that meaningful. And again, as you point out, there are a couple of potential ways to add that mass term in, either the "normal" way with a right handed neutrino, or with some fancy see-saw majorana term, or some combination thereof.
The issue is though, right now the standard model is at least ambiguous in terms of the majorana mass term. If it ends up the neutrino gets its mass from only the "normal" mass term, then why doesn't it have a majorana mass term? There's no current symmetry that says there can't be a majorana mass? If the neutrino's majorana mass is zero, then you'd probably have to introduce a symmetry into the standard model that says majorana particles can't exist.
But if the neutrino does end up having a non-zero majorana mass term then that means the neutrino is a majorana particle, and can undergo lepton number violating processes (e.g. neutrinoless double beta decay). Again, that's new physics.
So no matter how you give the neutrino mass, you're gonna have to modify the standard model in some "significant" way to accommodate. Either by specifically saying majorana particles can't exist, or by allowing for lepton number violating processes.
Now you could say, well then it might the case that majorana particles don't exist b/c that would require lepton number violating processes, so I don't need to introduce a new symmetry, I can just take advantage of one that's already lying around. That might be a valid claim to make...I'm not sure. I think the issue with that comes down to the difference between lepton number a global vs accidental symmetry in the standard model.
The issue is though, right now the standard model is at least ambiguous in terms of the majorana mass term. If it ends up the neutrino gets its mass from only the "normal" mass term, then why doesn't it have a majorana mass term? There's no current symmetry that says there can't be a majorana mass? If the neutrino's majorana mass is zero, then you'd probably have to introduce a symmetry into the standard model that says majorana particles can't exist.
But if the neutrino does end up having a non-zero majorana mass term then that means the neutrino is a majorana particle, and can undergo lepton number violating processes (e.g. neutrinoless double beta decay). Again, that's new physics.
So no matter how you give the neutrino mass, you're gonna have to modify the standard model in some "significant" way to accommodate. Either by specifically saying majorana particles can't exist, or by allowing for lepton number violating processes.
Now you could say, well then it might the case that majorana particles don't exist b/c that would require lepton number violating processes, so I don't need to introduce a new symmetry, I can just take advantage of one that's already lying around. That might be a valid claim to make...I'm not sure. I think the issue with that comes down to the difference between lepton number a global vs accidental symmetry in the standard model.