Does GR being locally Lorentz imply having to consider spinor irreps ($SL(2,mathbb{C})$) for consistence with its curved manifold tensor structure?

Even if not with the specific purpose of dealing with spinorial matter (like in supergravity, gauge formalisms, etc), does the mathematical consistence of GR entail that in a local (Lorentz) frame in the curved manifold of GR the Lorentz transformations associate the spinor transformations by $SL(2,mathbb{C})$ for observers in spacetime oriented orthonormal tetrads at an event $x$?

In other words, does making Lorentz symmetry local (unavoidable in presence of curvature and Lorentzian metric) bring about spinor fields associated to the usual tensor fields in GR when considering observers with measuring gauges at an event?

How are local sections of the orthonormal frame bundle at different points of spacetime uniquely related by local Lorentz transformations of the tetrads without using the spinor representation?