Edit: I know there have been some similar questions but I don't think any had quite articulated my particular confusion.
If gauge symmetries are really just redundancies in our description accounting for nonphysical degrees of freedom, then how does one explain the deep and powerful fact that if one begins with, say, just fermions and no gauge field in one's theory (no interactions & no dynamics), but then imposes that the theory be invariant under local U(1) transformations, then one finds a vector field must be introduced?
Note that I did not have any vector field in my theory before I demanded invariance under gauge symmetry. If one thinks of the vector field as being in the theory to begin with then I can see how one could see the imposition of local symmetry as a necessary constraint to remove extra degrees of freedom - you've got an A_mu, that thing's got 4 degrees of freedom and it should only have 2. But if I imagine that I knew nothing about photons or the electromagnetic field, and I require my theory of fermions to have this U(1) symmetry, then the vector potential arises as the mechanism for enforcing that symmetry. Beginning with no interactions, axiomatically or arbitrarily requiring gauge symmetry has the power to produce not only gauge fields in the theory, but the correct number of them and with the correct self-interactions (or lack of them).
I suppose one could say that SU(3) just happens to work because there just happen to be 8 gluons, similarly for SU(2) and U(1), but doesn't that seem awfully random and clunky (or... unnatural)? Doesn't it seem much more natural and coherent to say that there are 8 gluons precisely because there are 8 generators of SU(3), and so on? If I begin my theory without the gauge fields then it seems to me to make no sense to say that the shockingly powerful principle of requiring gauge invariance only accounts for a redundancy in a field that I have not even put in my theory yet and might not want to.
I cannot get away with imposing gauge invariance without introducing exactly the kind of forces and interactions that we observe and that appear in the SM. That statement seems way too powerful for a mere redundancy in our description. Again, maybe it's true that if you go about it from the other direction, ie, requiring one photon and three weak gauge bosons, etc., then you are forced to introduce the right gauge symmetry to account for the redundancies. But that seems much more ad-hoc to me - you have a lot of random things that happen to be true and a lot of coincidences that happen to work out - whereas if you think about the requirement of gauge symmetry as giving rise to these connections that tell you how to move around in your bundle, that sort of communicate the local transformation from one place to another, then you are only making one ad-hoc postulate, and it is a concise and elegant one with a ridiculous amount of explanatory power. It also seems weird to call the very thing that defines a theory only a dumb redundancy in the theory.
So, where am I wrong in all this? Is this an untenable view in the context of gauge theory (neglecting ideas that are being researched that may or may not pan out)? Or is this a viable view to take, even if one you dislike it? I know very little about spinor-helicity methods but I gather they may have something to say about this. Does their success eliminate the possibility of my interpretation?