What is the simplest physical argument to claim that Proca theory (involving a massive spin-1 boson) is not renormalizable?
This argument is going to be challenging, because the theory of a single massive vector interacting like a photon with scalar and spinor matter is renormalizable.
This is an accident of the U(1) gauge theory. It has a Stueckelberg version of the Higgs mechanism, where you take the limit that the Higgs mass goes to infinity while the condensate charge goes to zero. The limit is described on Wikipedia in the page on Higgs mechanism, under Affine Higgs mechanism.
This is an accidental property of massive electrodynamics. The reason the massive gauge field is nonrenormalizablei in general is that for a gauge group other than U(1), the charge cannot be arbitrarily small, so there is no decoupling limit for the Higgs.
The standard argument against a massive vector is that the propagator has a kk/ m2 term in the numerator iwhich means that longitudinal components don't have a falling off propagator, which means that loops involving longitudinal vectors blow up as a power at high k. In gauge theories, gauge inavariance guarantees that longitudinal bosoms aren't produced, but this requires that gauge invariance is broken spontaneously not by the Lagrangian. The Stueckelberg limit means that a mass term doesn't wreck renormalizability for U(1).