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What restrictions do you place on the G? For example let $G = Spin \times_{Pin^-} P(Pin^-)$ where $P(H)$ is the free path space on $H$ with pointwise multiplication. The fiber product uses restriction to one end of a path. Restriction to the other end of the path gives a surjective homomorphism from $G$ to $Pin^-$. But $G \simeq Spin$ and so $G$-bordism is the same as $Spin$-bordism.
@ManuelBärenz A PL-framing of S^7 includes (up to contractible choices) the data of a smooth structure and a smooth framing of the resulting (exotic) smooth S^7. You can think of it this way: a PL-framing is a lift of the tangent microbundle map from BPL(7) to a contractible space (e.g. EPL(7)). A smooth structure is a lift to from BPL(7) to BO(7), and a smooth framing is a further lift to a contractible space. So they are the same! PL-framed $S^7$s corresponding to distinct framed exotic $S^7$s are already distinct in the PL-framed bordism category (since the cats are equivalent).
I think being a flat bundle is a stronger property than being pulled back from $P_{\leq 1}X$. For example take $X$ to be the torus (or any genus $g$ surface with $g\geq1$). Then $X = P_{\leq 1}X$ agrees with its Postnikov truncation. So every vector bundle is a pulled-back bundle. However line bundles corresponding to non-trivial elements of $H^2(X, \mathbb{Z}) =\mathbb{Z}$ will have non-trivial first Chern class, and hence are not flat.
When $M=S^n$ is a sphere, then the space of orientation preserving homeomorphism is arc connected. In this case $G \to \textrm{Homeo}^+(M)/\textrm{homotopy} = pt$ seems to be no data at all? Is that really what you mean?
A related question: in what ways are non-comm. rings ($E_1$-algs) useful to an algebraic geometer? While there are places that non-comm rings appear (Brauer groups come to mind) an argument can be made that the field of non-comm rings is mostly concerned with things disjoint from the concerns of algebraic geometry. The $E_k$-operads interpolate between assoc ($E_1$) to "comm" ($E_\infty$), so you should expect a similar relationship. There are places where they show up (analogs of Brauer groups, the Deligne conjecture, deformations, etc), but mostly they are separate from the core concerns.
If C=Sets and T is the monoid for pointed sets, and $A=(A,a_0)$, then the Bar construction, as a simplicial set, is the union of $A \times \Delta[0]$ (i.e. const. simp. set on A) with $\Delta[1]$ along $\Delta[0]$ identified as $a_0 \in A$. It is not Kan, but it is a quasicategory.