I've been self studying Riemannian Geometry through Spivak's and Lee's books, and fairly often I've seen an argument that goes somewhat like this:
We have an operator that acts on vectors in the tangent space and, in order to show that it "lives at points", we generalise it to operate on global sections in vector bundles. Then, by showing this operator is linear over $C^\infty$ functions, some argument involving bump functions implies the operator defines a tensor field.
Usually this is followed by some remark about modules, giving a strong impression that something bigger is going on.
On general relativity texts, there's an equivalent practice that involves showing the operator "transforms correctly".
This is all quite vague, since I cannot quite point my finger at what's the general phenomenon. So my question is:
What is the relation between tensor fields and $C^\infty$ linearity and what does this has to do with modules?
First note that every vector bundle $E$ gives you $C^{\infty}(M)$-Module structure on global sections $\Gamma(E).$
Additionaly for vector bundles $E,F$ on maniflod $M$ there are $C^{\infty}(M)$-Module isomorphism $$\Gamma(E)\otimes_{C^{\infty}(M)}\Gamma(F)\simeq\Gamma(E\otimes_{\mathbb{R}} F)\hspace{10pt}\text{and}\hspace{10pt}\Gamma(E^{*})\simeq\Gamma(E)^{\vee}$$ where $E\otimes_{\mathbb{R}} F$ is bundle with tensor product taken fiberwise and $E^*$ is bundle with dual vector space taken fiberwise.
Now any tensor field is a section of bundle $TM\otimes\dots\otimes TM\otimes T^*M\dots\otimes T^*M.$ From previous you get that any vetor field can be treated as element of $$\Gamma(TM)\otimes\dots\otimes \Gamma(TM)\otimes \Gamma(TM)^{\vee}\dots\otimes \Gamma(TM)^{\vee}.$$ There are also isomorphism for symmetric and screw symmetric tensors. Namely $$\Gamma(E\odot E)\simeq\Gamma(E)\odot_{C^{\infty}(M)}\Gamma(E)\hspace{5pt}\text{and}\hspace{5pt}\Gamma(E\wedge E)\simeq\Gamma(E)\wedge_{C^{\infty}(M)}\Gamma(E).$$ Last, but not least due to the famous Swan's theorem you have that for any vector bundle $E$ $$\Gamma(E)^{\vee\vee}\simeq\Gamma(E).$$ It allows you to treat ${C^{\infty}(M)}$ linear functionals on $\Gamma(TM)^{\vee}$ as elements of $\Gamma(TM).$
For reference see:
Jet Nestruev, Smooth manifolds and observables, 2003
Lawrence Conlon, Differentiable Manifolds SE, 2001