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I kind of disagree on your statement about lepton violation. The standard model predicts lepton violating processes (sphalerons). The true symmetry of the standard model is B-L. Of course you are right, that these topological effects will not lead to majorana mass terms.

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Well, while sphalerons theoretically break B+L in the non-perturbative regime, they are exponentially suppressed at the energy level of our colliders. At the same time, irrelevant operators that also violate it are suppressed by the GUT scale. So even if you take the minimal Standard Model at face value, you're out of luck finding any sign of the violation either way. But if Neutrinos get Majorana masses, that would be an additional explicit violation at the perturbative level. That would be something we can directly observe, as in neutrinoless double beta decay.

I agree, that neutrinoless double beta decay would be incredibly interesting, but it is very speculative (and depending on the neutrino mass hierarchy not really falsifiable).

My original point was just that lepton number is not a good symmetry as it is broken by rhe chiral anomaly, which is not speculative at all. Of course, the sphaleron effects are negligible in collider settings, but for cosmology they are crucial and might be indirectly observable.

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