If gauge theory is a redundancy in our description of nature, how can there be phenomena which cannot be described without gauge theory? As I understand it, gauge "symmetry" can be seen as redundancy in our description of nature. For example, quoting from David Tong's gauge theory notes,

Gauge symmetry is,  in many ways,  an odd foundation on which to build our best theories of physics.   It is not a property of Nature, but rather a property of how we choose to describe Nature.   Gauge symmetry is, at heart,  a redundancy in our description of the world.  Yet it is a redundancy that has enormous utility, and brings a subtlety and richness to those theories that enjoy it.

So, in principle, one might think that one could do away with gauge theory, even if that might come at a considerable practical cost.
Yet, if I understand Chap 5.9 of Weinberg Vol. 1 correctly, it is impossible to write down a theory of a massless particle of helicity $\pm 1$ as an ordinary 4-vector field, and instead one must use a field which has a "gauge-like" transformation behaviour. (This is also discussed in this PSE post.) Thus, if we want to describe something like a photon, we are forced towards gauge theory.
Naively, these two ideas seem to contradict each other. How can we reconcile them?
 A: 
It is impossible to write down a theory of a massless particle of helicity $\pm 1$ as an ordinary 4-vector field, and instead one must use a field which has a "gauge-like" transformation behaviour. Thus, if we want to describe something like a photon, we are forced towards gauge theory.

You can describe the quantum theory of photons without gauge symmetry; this is literally Weinberg's starting point. What is more difficult is doing what Schwartz calls "embedding particles into fields". That is, defining local operators which create and destroy the particles, with certain nice Lorentz transformation properties. 
It's desirable to do this, even though it adds an extra layer of structure. For example, the fields transform more nicely than the particle states, and it is easier to see your theory is local if you construct it out of local fields. It isn't logically necessary, just practically necessary.
Furthermore, it becomes possible to define fields without gauge redundancy if you're willing to give up certain assumptions. For example, in atomic physics one usually works with a complete gauge fixing, leaving no gauge redundancy. This comes at the cost of tossing out Lorentz invariance. 
