The standard story goes as follows: gauge bosons cannot have a mass term because it would break gauge invariance in the lagrangian. This is clear, but why can't we just have massive vector bosons without any gauge transformations associated to them? My question arises from the standard model: spontaneous symmetry breaking gives you 3 massive bosons and a massless one, with a $U(1)$ residual gauge invariance from the subgroup of $SU(2) \times U(1)$ which leaves the $\varphi$ (vacuum) doublet invariant. My question is the following: why don't we just add massive vector fields with the desired masses and charges, without even defining gauge transformations for them?
On chapter 21 of Peskin&Schroeder they show how to quantize a spontaneously broken $U(1)$ theory in the $R_\xi$ gauge, and one interesting feature is that since the $\xi$ dependence drops out once you add both the (now massive) $A_\mu$ and the Goldstone boson $\varphi$ (with fake $\xi$ dependent mass) contributions, one can choose $\xi$ in different ways to prove different things, e.g. renormalizability, unitarity, and so on. Perhaps the theory wouldn't be renormalizable if we just added mass terms for the "non gauge" bosons, thereby making them fields with a Proca-like action.
