imagine that there is only gravitational force in the Universe and the matter has only gravitational mass("charge"). Accelerating any given piece of matter with respect to all the other matter would cause propagation of change (ie. wave) in the gravitational field of the whole system (Universe). Wave is energy and to generate wave a work is required, ie. a force has to be applied to the piece of matter being accelerated - the requirement for that force is what we call inertia.

Now of course the question - what is the carrier of that "gravitational charge". It would be great if Higgs boson was the carrier of that charge.

I thought the idea that "the Higgs boson is the field responsible for imparting mass" was the dominant theory, are there reasonable alternative explanations?

I don't think that's right. I think the Higgs creates a large amount of mass (90%-ish), but not 100% of it. (This is from memory of something I read, so I can't point you to a source...)