Let $(M,g)$ be an oriented Riemannian manifold of dimension $n$, and denote by $P_{\mathrm{SO}}\to M$ its oriented frame bundle. The usual definition of a spin structure is the data of a principal $\mathrm{Spin}(n)$ bundle $P_{\mathrm{spin}}$ together with a bundle morphism $P_{\mathrm{spin}}\to P_{\mathrm{SO}}$ such that a certain diagram commutes (the actions of the Lie groups on the principal bundles must be compatible with the bundle morphism). We say that a (Riemannian oriented) manifold is spin if it admits such a spin structure.
Of course it is well-known that if a manifold is spin, then the spin structure is essentially unique, and that the existence is a topological property (vanishing of the second Stiefel–Whitney class). In particular, it does not involve the Riemannian structure, and many applications in index theory implicitly carry a ‘this is in fact independent of the Riemannian structure’ subtext.
Let $E\to M$ be the fibre bundle over $M$ such that $E_x$ is the complete oriented flag manifold of $T_xM$, i.e. a point in $E_x$ is a increasing sequence of oriented subspaces of dimensions 1 to $n$. The model fibre $F$ is the complete oriented flag manifold of $\mathbb R^n$. Then it is my understanding that $E$ is canonically isomorphic to $P_{\mathrm{SO}}$, at least as a fibre bundle. In the same way that $\mathrm{SO}(n)$ admits a unique connected double cover $\mathrm{Spin}(n)$, $F$ admits a unique connected double cover $F'$, and we can ask if there exists a fibre bundle $E'\to M$ with model fibre $F'$ together with a bundle morphism $E'\to E$ such that $E'_x\to E_x$ is conjugate to $F'\to F$ for all $x\in M$.
The question of existence as it stands makes no reference to the Riemannian structure. If $(M,g)$ admits a spin structure, then of course forgetting the structure of principal bundle of $P_{\mathrm{SO}}\to M$ gives such an $E'$. I would bet that the existence of $E'$ ensures that the manifold is spin. In fact, I would take my chances saying that it gives, for any Riemannian metric on $M$, a canonical structure of $\mathrm{Spin}(n)$-bundle on $E'$ that makes $E'\to E$ into a spin structure (once $E$ is canonically identified with $P_{\mathrm{SO}}$).
Question
Is my understanding correct? In other words, is the loss of information due to the actions of $\mathrm{Spin}(n)$ and $\mathrm{SO}(n)$ irrelevant in purely topological terms?
Is this approach documented? Are there advantages to it, for instance does it smooth out some ‘this is in fact independent of the Riemannian structure’ arguments?
As far as I know, there is no natural group structure on $F$, and obviously we like group structures and principal bundles. In particular, many constructions of adjoint bundles are relevant to index theory. Short of being a principal bundle, is $E$ an adjoint bundle for some group, with an action that does not depend on the metric structure? In that case, we could maybe construct, say, spinor bundles as adjoint bundles with respect to this group.
