Why has the Higgs-field inertia?

The mass of the Higgs is not due to the Higgs mechanism. There simply is a mass-term in the Higgs field Lagrangian.

The whole idea that Higgs is the originator of mass of otherwise massless particles is quite mistaken, at least as it stands in the collective psyche of the popular culture. ;) The correct statement is that coupling to Higgs gives mass to fermions of the Standard Model and they would have been massless otherwise. As such, in the framework of quantum field theory, a field can simply have a mass term -- there is nothing wrong with it. However, when a theory has certain symmetries, it can forbid the field from having the mass because the mass term might violate the symmetry. This is precisely what happens with the fermions of the Standard Model, the $SU(2)_L$ gauge symmetry forbids the mass term. However, due to the coupling of these fermions with the Higgs field, via the mechanism of spontaneous symmetry breaking, they acquire a mass.

In any case, the moral is that there is nothing that says that you need to somehow have a source of mass. The Lagrangian of a field can simply have a mass term. It is the particular case of some field theories where certain particles might not have mass due to symmetries but they nonetheless acquire mass due to spontaneous symmetry breaking. However, this need not be the case. Ironically, as I mentioned earlier, the mass of the Higgs does not come from its coupling with itself. It is massive regardless of spontaneous symmetry breaking.