It is simply not true that "in the Standard Model, neutrinos are assumed to be massless, and the right-handed neutrinos thus do not exist".

When the SM was written down, no neutrino masses were known, so right-handed neutrinos were not included in it. Nevertheless, the electron did get a mass by a suitable higgs coupling to the left-handed electron multiplied by a right-handed SU(2)L-singlet electron. This is the celebrated "second job of the higgs", and theorists cheered when they appreciated its genius [ https://physics.stackexchange.com/questions/409313/higgs-couplings-and-fermion-masses/409414#409414 ].

As a result, instantly in 1968, any theorist would reassure you that, ipso facto, the neutrino /could/ easily get a mass with a very analogous coupling of the neutrino to a conjugate higgs doublet, if there were a reason to accommodate such. (In fact, in the extension to quarks, both up and down type quarks got their masses via SU(2)L-singlet right-chiral components.) They simply stuck to Occam's razor and left neutrino masses an open question, skipping them until discovery. It is doubly important to understand that /the discovery of neutrino masses did not upset or reverse absolutely anything about the structure and logic of the SM/.

Indeed, to hype up the surprising neutrino mass discovery, sensationalist self-promoting sides kept on insisting they went "beyond the SM", but this only betrayed they had never fully appreciated its symmetry structure and logic.

While the full nature of neutrino masses is still open, an SM Dirac mass for them analogous to the electron mass is not excluded.