That's an important point. That's why I stressed that the socalled "bare masses" of the elementary particles in the standard model come from the vacuum expectation value of the Higgs field (NOT from the Higgs boson, which is a certain state of the Higgs field).
The bulk (about 98-99%) of the mass of the matter around us is dynamically generated by the strong interaction and part of the somewhat enigmatic mechanism called "confinement". One way to at least check this theoretically is lattice QCD, i.e., numerically evaluating the hadron masses from QCD by calculating corresponding expectation values of quantum fields defined on a discretized space-time lattice and then taking the "continuum limit". This quite tricky business leads to a very good description of the known hadron spectrum, which makes us pretty confident that QCD works not only in the "asymptotic free limit" (i.e., for deep inelastic scattering), where perturbation theory is applicable, but also as a theory at low energies, describing not quarks and gluons but the hadrons, within which they are "confined".
From a symmetry point of view, there are in principle two mechanisms of mass generation. In the light-quark sector it's a good approximation to neglect the quark masses at all. In this massless limit of QCD, there's no length scale at all. The only dimensionful quantities are the fields themselves. If QCD were a classical field theory there'd be no way to get "mass without mass" (a famous saying by Wilczek), but first the theory is quantized, and the conformal symmetry of the classical massless-QCD Lagrangian is anomalously broken. In perturbation theory you formally see this that you have to renormalize the theory by subtracting divergences of Feynman diagrams forming closed loops. This subtraction leads to a redefinition of the coupling and the wave-function normalizations. However, you cannot subtract the corresponding one-particle irreducible diagrams (proper vertex functions) with all the external momenta vanishing since this introduces new divergences because all particles in the theory (quarks and gluons) are massless. You thus necessarily have to choose a scale ##\Lambda_{\text{QCD}}## and corresponding (space-like) momenta at this scale to subtract the divergencies. This introduces necessarily an energy scale and thus breaks the conformal symmetry. That leads to a non-vanishing trace of the energy-momentum tensor and that's why this anomaly (i.e., a symmetry of a classical field thwory which gets broken in the corresponding quantized field theory) is called trace anomaly.
Another way to generate "mass without mass" is hinted at by chiral symmetry. This refers to the fact that with vanishing quark masses the quark sector of QCD becomes chirally symmetric. Now the hadron spectrum (i.e., "QCD at low energies") doesn't show this chiral symmetry, because it implies that for each hadron there should be another hadron with the same mass but with opposite parity ("chiral partners"), but looking at the particles in the particle data book one always finds possible chiral partner states but with different masses. The explanation for this phenomenon is that chiral symmetry is spontaneously broken. Since this is a global symmetry (not a local gauge symmetry!) this implies the existence of massless states (Nambu-Goldstone bosons of the spontaneously broken symmetry). Now, there are no massless hadrons either, but there are the pions with a mass of about 140 MeV, which is pretty small compared to the usual hadron masses of around 1GeV=1000 MeV. This suggests that chiral symmetry is also a bit explicitly broken and thus the pions become a bit massive, but can be taken as approximate Nambu-Goldstone modes of broken chiral symmetry. Indeed it turns out that chiral perturbation theory is a successful effective theory of the hadrons as far as the light quarks (and also the strange quark) are concerned.
From a fundamental point of view, i.e., spoken in terms of quarks, lattice QCD shows that the spontaneous chiral symmetry breaking is due to the formation of a scalar "quark condensate", i.e., because of a non-vanishing vacuum expectation value ##\langle \bar{q} q \rangle##. Both the trace anomaly and the quark condensate are thus the prime culprits for the generation of "mass without mass". For a nice review, see
Roberts, C.D. Few-Body Syst (2017) 58: 5.
https://doi.org/10.1007/s00601-016-1168-z
https://arxiv.org/abs/1606.03909
