The central importance of Higgs boson would be that the Higgs mechanism gives rest mass to fundamental particles. It seems like a very natural argument that fundamental particles need to be given rest mass by a field interaction because as something fundamental (or a perturbation in a field as its often described), it can't have any properties beyond what a QFT can explain through interactions with a field.
What I don't understand is the role of matter-energy in this story. If the interaction of a particle with a field gave that particle rest mass, it seems logical that the field in the vicinity of the particle is in an excited state, which, integrated over the area of influence, is exactly equal to the rest mass of the particle. Why wouldn't this happen with interactions with other fields? Why does the (baseline) interaction of fundamental particles with the Higgs field create mass while interactions with the electric field, for instance, won't? In fact, it seems like we have lots of fields that could account for particle mass. Are those all just zero net-energy fields? What makes the Higgs capable of this while the others are not?
On this point, if the electric field is mediated by the photon force carrier, then how can the electric field from a charged particle have anything less than infinite energy? I believe it has zero energy, but I lack the tools to argue this. Are the photons mediating an electrostatic interaction countable? Or do they exist in a less real sense? The magnetic field, on the other hand, obviously has energy as evidenced by the existence of inductors. The gravitational field is the strangest of all since gravitational flux is proportional to mass itself. Rest mass is proportional to interaction with the Higgs field, and mass interacts proportionally with the gravitational field, but yet the Higgs field is not the gravitational field. Can't mass come from any of the fields? Surely, if negative charge orbits closely to a positive charge, the system's mass is less due to a phenomenon associated with the electromagnetic field. Is this all the correct perspective, or close?