Background :
Quantum mechanics is formulated with Hilbert spaces, of which the rays (possibly satisfying superselection rules) correspond to physical states. The space of (pure) states is thus a projective space and the symmetries are modelled by projective representations of the corresponding groups.
Half-integer spin label the projective representations of the Poincaré group that can't be related to an actual linear representation: one has to use the appropriate central extension, that is the universal cover of the Poincaré group. There is then a correspondence between linear representations of the universal cover and projective representations of the symmetry group. The universal cover acts on a "spinor space" and the Poincaré group acts on its ray space.
But the observables emerge from the projective representation, that is the ray space. In other words, the observable are bilinear (should I say hermitian?) in the spinor fields, that live in the spinor space.
I am interested in classical spin particles in a curved spacetime $M$. There the Poincaré group is usually reduced to the proper Lorentz group, with its universal cover the $\operatorname{Spin}$ group (I omit the signature argument). To construct the spinor space, we consider the bundle $\operatorname{SO}(M)$ of positively oriented local frames of our spacetime, then we add the topological data of a "spin structure" that gives us the corresponding covering of "spinor frames" $\operatorname{Spin}(M)$.
In the same way that tensors are associated to the bundle of local frames by their rules of transformations under a change of frame, the spinor bundle is constructed as the associated bundle to the spinor frame bundle for the action of the Spin group on spinors : $\Sigma_M = \operatorname{Spin}(M)\times_{\operatorname{Spin}}\Sigma$.
Question:
I have yet to find a convincing argument as to why this is an appropriate generalisation of the case of the Minkowski space in which spinor appear as projective representation of the global isometry group (see my question here). But assuming it is the right way to go, my question is where does the spin structure matter?
If the space of states is the ray space of sections of the spinor bundle $\mathbb{P}\Gamma(\Sigma_M)$ for some appropriate notion of sections (e.g. smooth, or some $\mathrm{L}^2$-completion), then it justifies why we need this $\Sigma_M$ bundle. But looking at the Lagrangian for a free spin-$\frac12$ particle: $$\require{cancel} L = \frac{i}{2} \bar{\psi} \stackrel{\leftrightarrow}{\cancel{\nabla}} \psi + m\bar\psi \psi$$ it only depends $\psi\otimes\bar\psi$ : that $i\!\!\!\stackrel{\leftrightarrow}{\cancel{\nabla}}$ can be interpreted as the Dirac operator acting on the tensor product $\bar\Sigma_M\otimes\Sigma_M$ (then contracted down to a scalar). But this $\bar\Sigma_M\otimes\Sigma_M$ is a bundle associated to the frame bundle $\operatorname{SO}(M)$. It has more information than just $\mathbb{P}\Sigma_M$ but its section have less information than the rays of $\mathbb{P}\Gamma(\Sigma_M)$ (the relative phase between different points is lost, but the norm data is preserved).
Hence it seems to me that at least in this simple example, the spin structure is not relevant and that the spinor bundle might just be a convenient way to represent the spin observables, yet its existence imposes topological constraints on the spacetime manifold. Am I missing something? Maybe there are some deeper motivations coming from QFT?
