Giving mass to neutrinos is not a problem. The problem is to explain its smallness. Once right-handed neutrinos $\nu_R$'s are included in the Standard Model (SM), the Majorana mass for $\nu_R$ must also be included unless that is forbidden by some symmetry. This automatically and unavoidably leads to the robust conclusion that the mass eigenstates (both light and heavy) are Majorana type.
So my question is what mechanism will generate Dirac neutrinos?
I am not entirely sure but I think without introducing new symmetries to the SM this is not possible and still then it is not quite natural. So I'll just summarize my ideas on this and see if someone can come up with something better.
To have a Dirac mass term you need to have some right handed neutrino $\nu_R$. As you said the general approach within the seesaw type 1 is to introduce a right handed singlet $\nu_R$ which gives you both a Dirac and Majorana mass term $$\mathcal{L}_m=\overline{L} \Phi\nu_R+\frac{1} {2} m_R\nu_R^TC\nu_R$$ Now since the Majorana mass term is only allowed due to $\nu_R$ being a singlet in the SM framework we need to change this to forbid this term.
A possibility to do so would be to just give $\nu_R$ a hyper charge so that the Majorana term is forbidden. All we need to do then is to introduce a second Higgs field with the corresponding hyper charge to allow for the Dirac term to appear. However this is not really a mechanism to produce massive neutrinos since the hyper charge would lead to a non-vanishing electric charge of the right handed neutrino and the additional Higgs. Neutrinos are neutral particles so this does not work and would have been seen in experiments. Also giving the new Higgs a non vanishing vev will break the electromagnetic ${\rm U(1)}$.
Therefore we need to introduce new symmetries to the theory to forbid the Majorana term to appear. Two possibilities are promoting the accidental lepton number symmetry to a ${\rm U(1)}$ gauge symmetry or introducing an additional ${\rm SU(2)_R}$ acting similar to its left-handed counterpart such that the Higgs transforms as $$\Phi\rightarrow U\Phi V^\dagger$$ for $U\in {\rm SU(2)_L}$ and $V\in {\rm SU(2)_R}$.
Interestingly the latter case necessarily comes with right handed neutrinos. Both symmetries would allow Dirac mass terms while forbidding the Majorana term. However both would probably need to be broken to hide the otherwise appearing massless gauge bosons. This breaking will again introduce Majorana terms for neutrinos.
One could avoid the breaking by assigning very small couplings to the new symmetries but this is a rather unnatural way and would call for some mechanism to explain the smallness of the couplings.
So in summary to my knowledge the only way to make neutrinos pure Dirac particles is to introduce new symmetries. One can however argue if we really need to speak of Majorana neutrinos in the case that the Majorana term is vanishingly small.
