There is a bicategory of Dixmier-Douady bundles of algebras which is equivalent to the bicategory of bundle gerbes. In particular, sections into these bundles form algebras.
The price you pay is that the bundles are infinite-dimensional; for that reson I am not sure if that picture persists in a setting "with connections".
I do not know a good source for the bicategory of Dixmier-Douady bundles or for the equivalence. Everything depends certainly on the type of morphisms you consider between the bundles; they clearly have to be of some Morita flavor. You may look into Meinrenken's "Twisted K-homology and group-valued moment maps", Section 2.1.1 and 2.1.4. In Section 2.4 Meinrenken indicates indirectly that his bicategory of Dixmier-Douady bundles is equivalent to the one of bundle gerbes, by transfering the notion of a multiplicative bundle gerbe (which depends on the definitions of 1-morphisms and 2-morphisms) into his language.
Side remark: a bundle gerbe is not the direct generalization of transition functions of a bundle. There is one step in between, namely a bundle 0-gerbe: instead of open sets, it allows for a general surjective submersion as the support for its transition functions. If you take bundle 0-gerbes instead of transition functions, the functor you mentioned at the beginning of your question has as canonical inverse functor. See my paper with Thomas Nikolaus "Four equivalent versions of non-abelian gerbes".
Added (after thinking a bit more about the question): If you want to categorify the vector space of sections into a vector bundle, you first have to fix a categorification of a vector space. An algebra is one possible version of a "2-vector space", probably due to Lurie. Another version, due to Kapranov-Voevodsky, is to define a 2-vector space as a module category over the monoidal category of vector spaces (add some adjectives if you like).
Let us define a section of a bundle gerbe $\mathcal{G}$ over $M$ to be a 1-morphism $s: \mathcal{I} \to \mathcal{G}$, where $\mathcal{I}$ is the trivial bundle gerbe. Then, sections form a category, namely the Hom-category $Hom(\mathcal{I},\mathcal{G})$ of the bicategory of bundle gerbes (the one with the "more morphisms" defined in my paper which was mentioned in the question).
The category $Hom(\mathcal{I},\mathcal{G})$ of sections of $\mathcal{G}$ has naturally the structure of a module category over the monoidal category of vector bundles over $M$. Indeed, a vector bundle is the same as a 1-morphism between trivial gerbes, i.e. an object in $Hom(\mathcal{I},\mathcal{I})$. Under this identification, the module structure is given by composition:
$$
Hom(\mathcal{I},\mathcal{G}) \times Hom(\mathcal{I},\mathcal{I}) \to Hom(\mathcal{I},\mathcal{G}).
$$
The functor which regards a vector space as a trivial vector bundle induces the claimed module structure over vector spaces.
Summarizing, sections of bundle gerbes do not directly form algebras, but they form Kapranov-Voevodsky 2-vector spaces.
To see that the Hopf bundle is not trivial, one considers its restrictions to the subspaces $N,S\subset S^2 = \mathbf{CP}^1$, with $N = \mathbb{C}^*\cup\{\infty\}$ and $S = \mathbb{C}$ where it is trivial. One should be able to write down sections given this description. Then the transition function $\mathbb{C}^* \to S^1$ can be written down. You can think of this as a map $\mathbb{C}^* \to \mathbb{C}^\ast$, and hence calculate the integral of it around $S^1 \subset \mathbb{C}^*$. This gives you the index of the transition function, which is non-zero.
You can then calculate the index of any transition function for the Hopf bundle, using the fact that it will be a Cech cocycle equivalent to the one you've written down.
Lastly, you can calculate the index of the transition function for the trivial $S^1$-bundle on $S^2$, and find this is not equal to that for the Hopf bundle.
In fancier language, because $\mathbb{C}^*$ is not simply connected, one finds that you cannot deform the Hopf bundle's transition function, which is not null-homotopic, to the transition function for the trivial bundle, which is null-homotopic.
In pictures, one has that the transition function for the Hopf bundle, restricted to the circle, loops once around the origin, but the trivial bundle's transition function is constant. If people are happy with believing that continuously deforming the transition function gives equivalent bundles (one could motivate this by writing down Cech coboundaries on $\mathbb{CP}^1$ that give the equivalence), and that transition functions which cannot be deformed to each other give inequivalent bundles (this is the important point), then this is pretty much the best picture you'll get.
Best Answer
If $X$ is a manifold, and $E$ is a smooth vector bundle over $X$ (e.g. its tangent bundle), then $E$ is again a manifold. Thus working with bundles means that one doesn't have to leave the category of objects (manifolds) under study; one just considers manifold with certain extra structure (the bundle structure). This is a big advantage in the theory; it avoids introducing another class of objects (i.e. sheaves), and allows tools from the theory of manifolds to be applied directly to bundles too.
Here is a longer discussion, along somewhat different lines:
The historical impetus for using sheaves in algebraic geometry comes from the theory of several complex variables, and in that theory sheaves were introduced, along with cohomological techniques, because many important and non-trivial theorems can be stated as saying that certain sheaves are generated by their global sections, or have vanishing higher cohomology. (I am thinkin of Cartan's Theorem A and B, which have as consequences many earlier theorems in complex analysis.)
If you read Zariski's fantastic report on sheaves in algebraic geometry, from the 50s, you will see a discussion by a master geometer of how sheaves, and especially their cohomology, can be used as a tool to express, and generalize, earlier theorems in algebraic geometry. Again, the questions being addressed (e.g. the completeness of the linear systems of hyperplane sections) are about the existence of global sections, and/or vanishing of higher cohomology. (And these two usually go hand in hand; often one establishes existence results about global sections of one sheaf by showing that the higher cohomology of some related sheaf vanishes, and using a long exact cohomology sequence.)
These kinds of questions typically don't arise in differential geometry. All the sheaves that might be under consideration (i.e. sheaves of sections of smooth bundles) have global sections in abundance, due to the existence of partions of unity and related constructions.
There are difficult existence problems in differential geometry, to be sure: but these are very often problems in ODE or PDE, and cohomological methods are not what is required to solve them (or so it seems, based on current mathematical pratice). One place where a sheaf theoretic perspective can be useful is in the consideration of flat (i.e. curvature zero) Riemannian manifolds; the fact that the horizontal sections of a bundle with flat connection form a local system, which in turn determines the bundle with connection, is a useful one, which is well-expressed in sheaf theoretic language. But there are also plenty of ways to discuss this result without sheaf-theoretic language, and in any case, it is a fairly small part of differential geometry, since typically the curvature of a metric doesn't vanish, so that sheaf-theoretic methods don't seem to have much to say.
If you like, sheaf-theoretic methods are potentially useful for dealing with problems, especially linear ones, in which local existence is clear, but the objects are suffiently rigid that there can be global obstructions to patching local solutions.
In differential geomtery, it is often the local questions that are hard: they become difficult non-linear PDEs. The difficulties are not of the "patching local solutions" kind. There are difficult global questions too, e.g. the one solved by the Nash embedding theorem, but again, these are typically global problems of a very different type to those that are typically solved by sheaf-theoretic methods.