You're right – maybe I should have been more precise with my wording.

Let me address curt15's (and your) question in two ways:

####### The physicist's argument #######

In classic electrodynamics it doesn't make much sense to talk about a U(1) gauge symmetry (whether global or local) – because there's no electron field exhibiting such a symmetry in the first place. Gauge symmetries really only become important when talking about quantum mechanics, Dirac's equation and, more generally, coupling particle fields to force-carrying boson fields in a QFT. There it turns out that imposing local gauge invariance would guarantee you a way to accomplish the coupling while preserving the conservation of the Noether current associated with the global gauge symmetry.

But in classical electrodynamics, none of that is needed and the vector potential is simply a vector field on Minkowski space. In fact, the value of the vector field and the choice of gauge are not even important because there's no classic analogue of the Aharonov-Bohm effect.

Now pretty much the same applies to Einstein-Maxwell theory: There is no electron field and no U(1) gauge symmetry. It's all classical. Indeed, in my experience relativists just view the vector potential as a vector field on spacetime and ignore the QFT-inspired classical gauge theory stuff (principal bundles, associated vector bundles, etc.) altogether. You might disagree with this approach but given that no one has yet managed to write down any 4-dimensional, coupled quantum field theory in a mathematically coherent fashion (let alone on a curved background), I wouldn't say there's too much pressure to incorporate results from QED into GR.

Back to the paper: The authors basically start from the same perspective, i.e. the foundations of Special Relativity and classic electrodynamics where A was simply viewed as a vector field. They then propose an alternative to General Relativity and, in fact, electrodynamics(!) Like true physicists, they first worry about the functorial nature of their objects (what do they do & how do they relate to one another), not about the category the objects live in. If that means redefining[2] what A is in a mathematical sense, so be it. At the end of the day, experiments are what matters.

####### The mathematician's argument #######

Could it be that you and curt15 are thinking of the transformation behavior of A under local gauge transformations[0], not coordinate transformations? Because, unless I'm mistaken[1], the transformation of A under coordinate transformations is the same as for a vector fields, see definition 5.4.1 on p. 270 of [3]. The relevant portion reads:

Let p: P -> M be a principal G-bundle on the manifold M and A a connection 1-form on P, let s: U -> P be a local gauge of the principal bundle on an open subset U of M. Then we define the *local connection 1-form* (or *local gauge field*) A_s on U by

A_s := A ○ Ds = s* A

If we have a manifold chart on U and {∂_μ} (μ=1,…,n) are the local basis vector fields on U, we set

A_μ := A_s(∂_μ)

Now since the basis vector fields ∂_μ transform in the usual way and A_s is linear, the components A_μ should transform just like those of a section of the cotangent bundle T*M. Am I missing something?

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[0]: See the transformation formula in https://en.wikipedia.org/wiki/Gauge_theory#An_example:_Scala... or theorem 5.4.2 of [3].

[1]: It's been a while since I last did mathematical gauge theory, so please let me know in case I'm committing any errors here.

[2]: Since g is defined in terms of A, A must necessarily be a globally defined vector field and cannot be a local gauge field.

[3]: Mark J.D. Hamilton: "Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics" (Springer, 2017)