Spontaneous breaking of a (global or gauge) symmetry does not increase the number of degrees of freedom, it preserves it. When a spontaneous symmetry breaking happens, the system goes from a unstable vacuum to a true stable vacuum. This implies that what are the physical excitations around the vacuum change but it is simply a reorganization of the same degrees of freedom. This is obvious with a Lagrangian at the classical level: if a symmetry is broken by some expectation values of the fields then to understand the theory after the symmetry breaking, we have to expand the fields around these expectation values: it is just the same theory expanded around a different point so the number of degrees of freedom are the same.

In general, the reorganization of the degrees of freedom is non-trivial (example: Higgs mechanism for spontanously broken gauge symmetries, "absorption of the Goldstone bosons by the broken gauge bosons") and will deeply affect the RG flow: for example, in the Higgs mechanism, some gauge bosons become massive and so are integrated out in the IR whereas they would survive as massless particles in the unbroken theory.

The GUT example is not correct: if we start with a big gauge group which is broken by a Higgs mechanism to several small gauge groups, the number of degrees of freedom remains the same as argued before (they are more groups but they are smaller) and after pushing the RG flow and integrating the massive gauge bosons, the number of degrees of freedom decreases.

In fact, the a- and c-theorems apply without the assumption of no symmetry breaking. It is one of the greatest interest of this results. More precisely: in general, it is very difficult to know if a spontaneous symmetry breaking occurs in a strongly coupled QFT whereas this question is of central importance to understand the IR behavior of the theory. But the a-,c- type theorems give us some tools: one can say, let us assume that such symmetry is (or not) spontaneously broken then determine the IR theory with this assumption and compare it with the UV theory: if this violates the general a-,c-type inequalities, then this means that in fact the symmetry is not (or is) broken.