The claim is often made that the discovery of the Higgs boson will give us information about the origin of mass. However, the bare masses of the up and down quarks are only around 5 MeV, quite a bit smaller than their "constituent" or "dynamical" mass of around 300 MeV. (Remember that a neutron, for example, is one up and two down quarks and has a total mass of 939 MeV.) What then, is the reasoning behind the claim that the Higgs will address the origin of mass when by far the majority of the mass of the neutron (and proton) is related instead to the dynamical breaking of chiral symmetry?
The extra mass of the proton and neutron is not due to chiral symmetry breaking. It is due to the energy in the electromagnetic and strong force fields.
If chiral symmetry were an exact symmetry of the Lagrangian then the pions and other mesons (not the baryons) would have zero mass due to spontaneous chiral symmetry breaking. The chiral symmetry is not exact due to the small bare masses of the quarks so the mesons are not exactly massless.
From this you can see that your question is a bit confused, but one part that is correct is that most of the mass in ordinary atomic matter is not due to the Higgs mechanism. When people say that the Higgs boson will give us information abut the origin of mass they mean the bare masses of non-composite particles such as electrons and quarks.
Your point is totally valid and it is a favorite point of David Gross who happened to co-discover Quantum Chromodynamics. Indeed, it is often said by the journalists that the Higgs boson is the God particle that gives everything its mass. Without this mass, everything would be massless.
However, most of the mass of the visible matter is stored in the mass of protons and neutrons. And approximately 98% of the mass of these nucleons comes from the complicated QCD interactions inside these particles which also create lots of virtual gluons and quark-antiquark pairs, aside from large kinetic energy of all these particles that makes their total mass substantially exceed the rest mass. Only 2% or so of the nucleons' mass boils down to the Higgs-generated rest mass of the up and down quarks.
It is a bit subtle whether one should say that this 98% of the proton mass comes from the chiral symmetry breaking. It would be totally correct for the pions. Pions would be massless if the chiral symmetry were not broken; it would be a bit different with the nucleons because their mass would not be protected even if the chiral symmetry were exact.
However, there is a sense in which the God particle is necessary even for this dominant QCD-generated mass of the proton: you need nonzero quark masses to really break the chiral symmetry. The chiral symmetry breaking just amplifies this initial "impulse" - the nonzero rest masses - to break the chiral symmetry.
So of course, you and David Gross are right that the journalists exaggerate the importance of the God particle; most of the actual mass of the materials we know comes from the QCD mess. However, even the chiral symmetry breaking would be hard without the God particle.