Rest-mass being a somewhat difficult concept if your particle only "exists" in bound systems, i.e. is confined. But yeah, forget the Higgs, almost all the mass you see around you is generated dynamically by non-perturbative QCD.

Edit: More explanation: If you take the quark masses as unbound rest mass, it's not clear why the proton is bound -- it would have higher energy (=mass) than the unbound constituents. This is different from atoms, as a hydrogen atom has slightly less mass than the sum of a free proton and free electron (13.6eV in the ground state).

The quark mass is somewhat misleading: It's not the mass of a "free" quark, that doesn't exist, because of confinement. The incredible strong nuclear force actually /lowers/ the mass of the proton compared to the sum of the unbound constituents, from "infinity" to some finite value. Strange, I know, but similar to what happens with the hydrogen atom: In the ground state, it's 13.6 eV less massive than the sum of the mass of proton and electron.