Spontaneous Symmetry of a local gauge symmetry would be catastrophic as Elitzur's Theorem would be violated.
Your question is about the use of mathematics in physics and the physical meaning of said mathematics, specifically in the context of the Higgs Mechanism.
You are referring to the fact that local gauge symmetries are redundancies as opposed to physical symmetries. (Look into Quantum Gravity No Hair theorem for cool implications on global symmetries)
I suspect that concepts like BRST quantization and ghost fields would also fall in your "confusing" category, and I can see why that is.
Your choice of the Higgs mechanism is particularly interesting because you can think of it in a rather intuitive way. Let me give a brief overview.
What happens is that the Higgs field, from being an SU(2) doublet, acquires a vacuum expectation value, which is a fancy way of saying that you can expand your field about some non-zero minimum value. Thus far everything is physical. Now the idea is that through your covariant derivatives, and gauge freedom you can choose a specific gauge, in which this expansion (which through Goldstone's theorem has given rise to goldstone bosons, think of them as degrees of freedom) such that your theory looks like what we call mass. The use of gauge symmetries is the same as switching coordinate frames. If you consider moving charged particles and switch coordinates, you get different electric and magnetic fields, so it feels like a physical change appears just by expressing your theory from what should be a physically equivalent formulation, and in fact, it is, you just need to be careful with what you call, in our case, mass, and fields.
There is nothing unphysical about acquiring a VEV, or at least this specific description, or about EWSSB, or the mass acquisition. What is unphysical is the mathematical choice we make to express our terms in a nice and intuitive way.
