What goes wrong if we add a mass term for gauge bosons without the Higgs mechanism? 
Question: Why can't we add a mass term for the gauge bosons of a non-abelian gauge theory?

In an abelian gauge theory one can freely add a mass and, while this breaks gauge invariance, as long as the coupling current is conserved everything works fine (i.e., the scalar modes decouple and the theory is renormalisable).
In non-abelian gauge theories, it is often stated that the only way to introduce a mass term is through the Higgs mechanism. If we added a mass term without introducing the Higgs field, but the coupling current is still conserved, at what point would the theory break down? It seems to me that the scalar modes decouple as well, at least to tree level. I failed to push the calculation to one loop order, so maybe the theory breaks down here. Is this the most immediate source of problems, or is there any simpler observable which fails to be gauge invariant?
One would often hear that if we break gauge invariance the theory is no longer renormalisable. I may be too naïve but it seems to me that a (gauge-fixed) massive gauge boson has a $\mathcal O(p^{-2})$ propagator and therefore (as long as the current in the vertices is conserved) the theory is (power counting) renormalisable. Or is it?
To keep things focused, let us imagine that we wanted to give gluons mass, while keeping self-interactions and the coupling to matter (and ghosts) unchanged. Could this work without a Higgs?

There are many posts about that are asking similar things. For example,


*

*Can mass develop without the Higgs mechanism? asks about the mass of fermions, not of the gauge bosons.

*Massive Gauge Bosons without Higgs Effect same as above.

*Is Higgs mechanism necessary in QCD? same as above.

*What are the alternatives to the Higgs mechanism? is a list of modifications/generalisations to the Higgs mechanics, but doesn't discuss if it is avoidable at all.
 A: What a great question OP! I have good news and bad news. The good news is that this exact same question is asked and answered in Quantum Field Theory, by Itzykson & Zuber, section 12-5-2. The bad news is that the answer is
If you introduce mass terms in non-abelian gauge theories by hand, the theory is non-renormalisable.
This means that one is forced to introduce the Higgs mechanism (or variations thereof, such as the Stückelberg mechanism), which for some people is rather inelegant (and plagued by problems of naturalness, etc). Oh well, that's the way the cookie crumbles.
Let me quote the first paragraph of the aforementioned section, so as to summarise the main point of the problem:

Is a gauge theory where mass terms are introduced by hand renormalizable?
In electrodynamics, the situation is favorable. After separation of the gauge
field into transverse and longitudinal components, the longitudinal part $k_\mu k_\nu/M^2$ which gives rise to the bad behavior in the propagator does not contribute to the $S$ matrix. This results from the noninteraction of longitudinal and transverse components and from the coupling of the field to a conserved current. In a nonabelian theory, none of these properties is satisfied. Longitudinal and transverse parts do interact, while the current to which the gauge field is coupled is not conserved. On the other hand, unexpected cancellations of divergences at the one-loop level make the theory look like renormalizable. This explains why it took some time to reach a consensus, namely, that the theory is not renormalizable. The way out of this unpleasant situation is to appeal to the mechanism of spontaneous symmetry breaking, to be explained in the next subsection.

