Just to clear up several misconceptions at once:
1: The Higgs boson itself isn’t responsible for giving anything mass. That’s the Higgs field. The Higgs boson is just an excitation of the field, and detecting those excitations is evidence in favor of the Higgs mechanism, but it’s not directly involved in it. By way of analogy, we can think of the Higgs field as a trampoline, and assume that the trampoline is responsible for something important. If you thwack the trampoline in the right way, then you can make ripples in it (the ripples being analogous to the particle). If you see those ripples, then you know that the trampoline exists, but the ripples aren’t necessary for the trampoline’s existence.
2: The Higgs boson isn’t responsible for all mass. The Higgs mechanism applies only to fundamental particles, not composite ones, and probably not even to all fundamental particles (the neutrinos may well instead get their mass through a completely unrelated mechanism). So the Higgs isn’t responsible for mass as a whole, just some specific examples of mass.
3: The Higgs boson isn’t even responsible for most of the mass we’re familiar with. Most of our mass comes from protons and neutrons, and protons and neutrons are made up of quarks, and the masses of the quarks come from the Higgs process. But the mass of a proton or neutron is much, much higher than the sum of the masses of the three quarks that make it up: Most of the mass of a proton or neutron comes from the strong-force binding energy that’s holding those quarks together. And thus, most of our mass likewise comes from that strong-force binding energy.