I might agree if I knew what you meant.
The physical interpretation of that math would go like this: the "massless" elementary particles are coupled to the Higgs field, which "dresses" the particles in a cloak of virtual Higgs particles, and it is this cloak that plays the role of mass. The stronger the coupling, the heavier the cloak.
Hmm... I think I see what you are getting at. The Higgs mechanism is a mathematical concept that may be interpreted physically. Degrees of freedom would also be a mathematical concept.
Allow me to try to explain the difficulty I had with your original statement. It is not that I think it was incorrect; but rather, it reflects a way of looking at the world that is different from the way that I, as an engineer, view it.
Consider gravitational mass. Everyone has an idea, acquired by direct experience, that massive objects are attracted to the earth. Since Newton's time, we have written F = mg. But does this equation "explain" or "account for" gravitational mass? My contention is no: F = mg is a convenient mathematical description of what we observe to happen (at least in most everyday applications).
Put it another way, does nature obey our mathematics? Or do we hope our mathematics describes nature? My impression is the mathematician or mathematically inclined physicist considers the equations more real than nature. Thus, Goldstone bosons are not objects but degrees of freedom.
Well, you asked me a question about how the model behaves, and so my answer was of course written in the context of the model being correct, which it may not in fact be. As for my original statement, "the Higgs mechanism may explain why quarks and leptons have mass," what it means is that--take a flyer for a moment--assuming that SU(2) X U(1) is the correct symmetry to describe a unified electroweak force and assuming that it is spontaneously broken by the Higgs mechanism, lepton and quark masses follow as an unavoidable corollary. (Yeah, I know that didn't make sense to you on its surface, but pretend it's an opera: ignore the Italian and groove to the music.)
But you are asking a deeper question, now. The way I look at it is that nature is, at its core, consistent with mathematical truth. When you talk about "our mathematics", you mean, "our mathematical formalism", that is, our choice of symbols and our methods for manipulating them. Using "our mathematics", however, we do uncover real mathematical truths. These truths are universal, and they do constrain the possible behavior of reality.
Mathematical truth is more fundamental than reality; mathematical formalism is not. We attempt to use our mathematical formalism in such a way as to uncover those truths that are relevant to the description of reality. When we have a description of reality (say, a description of how particles acquire mass) then we can make ironclad predictions about how nature would behave in such a universe, and test to see whether our universe does, in fact, match those predictions. The Higgs boson is one such prediction.
Hah! You engineers! I'm still suffering confusion because the negative pole on my car battery has a big "+" sign on it.