Why do people say that the Higgs mechanism gives mass to the gauge bosons without mentioning the fermions?

The general problem that the Higgs mechanism solves is giving mass to spin-one particles. It turns out that finding relativistic, unitary theories of spin-one massive particles is non-trivial. There are a few known ways of doing it (this paper has a pretty good list of sources), but the oldest and easiest is probably the Higgs mechanism.

In contrast, there is no fundamental difficulty with theories with massive spin-1/2 particles. It's as easy as writing down the Dirac Lagrangian.

In the special case of the standard model we know that the fermions must couple to a chiral gauge field, and so the mass term must arise from spontaneous symmetry breaking, making the Higgs mechanism the answer to two problems instead of one. But in general, the Higgs mechanism is for giving mass to spin-one particles.

Of course the SM Higgs gives mass to both fermions as well as the gauge bosons. However, the latter is much more fundamental and predictive than the former.

Point is, it is enough for a scalar to transform non-trivially under a gauge symmetry to contribute to its associated gauge bosons masses (after taking vev). This contribution is constrained by the gauge sector and you do not have a lot of freedom. However, in order to contibute to the fermions masses, you need to be in a very specific representation and the couplings are, a priori, arbitrary.