> the standard model neutrinos can't participate in the Higgs mechanism due to always being left-handed

This again? It's only true if you insist on sticking with the original form of Weinberg's "model of leptons" from 1967 [1], which was written when massless neutrinos were consistent with available experimental data. Adding quark-style (i.e. Dirac) neutrino mass terms to the Standard Model is a trivial exercise. If doing so offends some prejudice of yours that right-handed neutrino can not exist because they have no electric and weak charge (in which case you must really hate photons too, not to mention gravity) you can resort to a Majorana mass term [2] instead.

That question (are neutrinos Dirac or Majorana?) is not a "contradiction", it's an uncertainty caused by how difficult it is to experimentally rule out either option. It is most certainly not "a problem for the standard model".

[1] https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.19.1264

[2] https://en.wikipedia.org/wiki/Majorana_equation#Mass_term