I have a question concerning fermions in curved space-time. Please read it to the end before suggesting the spin-connection and vierbein-based approach.
I heard that there is a special way of thinking about spin-1/2 particles (Dirac fermions) in flat space-time: the spinor field $\psi(x)$ is considered a (Grassmanian) scalar multiplet (under the Lorentz transformations), but the matrix-valued 4-vector $\gamma^{\mu}$ transforms as an actual 4-vector.
The value of the $\psi$ field here is in correspondence with the value of the usual spinor-transforming field, but taken at some fixed frame of reference (in which $\gamma^{\mu}$ take the usual fixed values). Quantities like $\bar{\psi} \gamma^{\mu} \psi$ transform like vectors, which is basically why this formalism is equivalent to the standard (with $\psi$ transforming as spinor and constant $\gamma^{\mu}$).
The Dirac action is then just $$ S[\psi] = \int d^4 x \: \bar{\psi} \left( i \gamma^{\mu} \partial_{\mu} - m \right) \psi, $$ which is manifestly Lorentz-invariant in this strange formalism.
My question is about curved space-time of GR. The idea is to write something like $$ S[\psi] = \int d^4 x \: \sqrt{-g} \: \bar{\psi} \left( i \gamma^{\mu} \partial_{\mu} - m \right) \psi, $$ where $\gamma^{\mu}$ transforms as matrix-valued vector under GCTs, $\nabla_{\mu} \psi$ and $\partial_{\mu} \psi$ are equivalent since $\psi$ is basically a (Grassmanian) scalar multiplet. So this new action is manifestly diffeomorphism-invariant, and agrees with the Dirac field in flat space limit. Also (since $\left\{ \gamma^{\mu}, \gamma^{\nu} \right\} = 2 g^{\mu \nu} \cdot 1_{4 \times 4}$) the metric field can be constructed out of (more fundamental?) matrix-valued vector field $\gamma^{\mu}$.
My teacher says it is incorrect, and I am pretty sure it is, but he can't explain why (and that's what really bothers me). One guess is that the interaction between fermions and gravity probably not correct since there is no spin-connection term (like in the standard vierbein-based approach).
So the question then becomes: what should I add in this action to make the fermion-gravity interaction term correct, given that I don't want to abandon this strange formalism and consider the spinor transformation of $\psi$.