[cygx]

Higgs boson == inertial mass is very misleading.

Mass is essentially rest energy by another name. The presence of a non-zero Higgs field gives certain elementary particles that would otherwise be massless such a non-zero rest energy.

In contrast, compound 'particles' (hadrons, atoms, chairs and tables, ...) only gain a miniscule amount of mass from the Higgs mechanism: They are bound states of interacting constituents that are whizzing around, generating ripples in various quantum fields (sometimes described as clouds of virtual particles), with the biggest contribution by ripples in the field of the strong nuclear force.

Now, in quantum theory, any field comes with associated particles, and for the Higgs field these are the Higgs bosons, and for the gravitational field these are the (conjectured) gravitons.

While gravitons would be intimately related to how gravity works at the quantum level, the relation of Higgs bosons to inertia is rather incidental.