I'm not too familiar with the physics side of things, so maybe I'm misunderstanding the notion of gauge symmetry, but even in the absence of matter I think there is a U(1) gauge symmetry present. The homogeneous Maxwell's equations are expressed purely in terms of the curvature F of the connection, if I remember correctly. So you might add a flat U(1)-connection to A without changing the equations of motion.

As for the math: the key point is that the metric is globally defined on the spacetime manifold M. I agree that the A_\mu transform as the coefficients of a differential form (A is a connection after all), but the notation elides the fact that the A_\mu are defined only on U. They depend, in particular, on the choice of local gauge s:U \to P. So the question about covariance (or globalization or coordinate-independence or whatever you want to call it) of both sides of the equation g_{\mu\nu}=A_\mu A\nu very much involves the question of local gauge transformations.

Either the principal bundle is assumed to be trivial, in which case there exists a global gauge and the connection A can be identified with a global 1-form on M (connections form an affine space modeled on such 1-forms; a gauge effectively converts this affine space to a vector space by choosing an origin), or we need to check that the right-hand side A_\mu A_\nu is indeed a symmetric two-tensor on M. The latter is not clear to me.

I admit I haven't looked at the paper carefully, and physicists typically don't approach things in such an explicitly mathematical way. So perhaps there's some (physical?) justification for why that equation typechecks. I don't quite see it though.