Is the discovery of the Higgs boson tantamount to discovering the Higgs mechanism? I mean, it's clear by now that the Higgs boson is there, but what about the Higgs mechanism? How can we ever be sure the mechanism is really at work? The whole theoretical framework did lead to the prediction of the boson, but could it be that it was accidentally discovered, without the framework being there? Say within the framework of another theory that doesn't require a Higgs field to be present to give mass to particles?
No, the discovery of the Higgs boson has little to do with validating the Higgs mechanism, or the breathtaking role of the Higgs field Yukawa couplings giving masses to fermions.
There are several "Higgsless models" accommodating the Higgs mechanism just fine; and a very heavy and broad (and hence barely visible, σ-like) Higgs would have little to say on the nature of the Higgs mechanism; it would only intrigue those that are concerned with naturalness, and do not consider the quartic Higgs model as an effective field theory.
"We" have trusted the SSB & Higgs mechanism is behind the masses of both gauge bosons and fermions since the triumph of the Standard Model in the 1970s. Observing the Higgs particle simply validated the simple quartic renormalizable potential picture on the margins of the model--it was the only model-dependent part of the firmly established theory of Electroweak unification of Glashow, Weinberg, and Salam, relying on a Higgs doublet field.
Specifically, the masses and telltale couplings to fermions of the SM model were being relentlessly confirmed for 40 years before the actual "remnant" Higgs boson itself was discovered. The entire symmetry structure of SSB as specified by the model was explored in neutrino scattering, and, in general, systematic phenomenology of weak decays and interactions, in which the Higgs particle itself features very little. A look at the outstanding summary of the electroweak theory in the PDG should confirm that by inspection.
The key point is that there hardly is any viable model, let alone theory, that can do the job of the three Goldstone degrees of freedom involved in the masses of the Ws and the Z, and Yukawa-coupled to the fermions so as to give them the chirally consistent masses so admired.
Obtaining the astoundingly accurate constraints of couplings to fermions is a "sudoku" solution that still baffles serious thinkers of such models by its minimal elegance. Much of the work on GUTs is an effort to understand it. (I watched Feynman struggling to see the method in the SM's "cockeyed" (his word) madness. I do not recall how much he traced them to GUT models, but certainly Gell-Mann and collaborators did try, intriguingly.)
The takeaway is that the electroweak theory has been firmly king of the roost for virtually half a century now, whereas our millennium's study of the Higgs particle couplings and properties, important as it is, is unlikely to shed much light on the above, unless a smoking-gun discrepancy were discovered tearing down the established magnificent big picture.