I would argue that the Higgs mechanism is just that: A mechanism for explaining where mass comes from. You could explain charge in a similar way by saying “because the particles are made of a certain amount of up or down quarks”.

Neither of these explanations answer the underlying question “but why does the Higgs mechanism give things mass?” or “but why do up/down quarks give things charge?”.

My point is that, at some stage, you get to the point of “the Higgs boson has mass because it’s an intrinsic property of the Higgs boson”, which is tantamount to “they just do”.