I have seen some statements that the Higgs boson is not responsible for particles gaining mass, but rather the Higgs field is. However, as I understand it, the Higgs boson is just an excitation in the Higgs field just as a photon is an excitation in the electromagnetic field. Additionally, the differences between the right chiral and left chiral particles of each type seem to me to exactly match the properties of the Higgs boson (weak hypercharge of 1, weak isospin of -1/2). Is it wrong to think of the interaction between the two chiral components as an exchange of virtual Higgs bosons in the same way the virtual gauge bosons can represent the interactions with the electromagnetic/weak/strong forces?
Is it wrong to think of the interaction between the two chiral components as an exchange of virtual Higgs bosons in the same way the virtual gauge bosons can represent the interactions with the electromagnetic/weak/strong forces?
Indeed, it is quite wrong. No particle exchange is responsible for the mass of the fermions. In fact, no excitation (like the Higgs particle) is!
(You are right that the Higgs particle is linked to these masses, as opposed to the other three degrees of freedom/components of the Higgs field, responsible for the masses of gauge bosons, and the Higgs mechanism.)
However, for fermion masses, the invariant Yukawa couplings to fermions involve a field which has a non-vanishing expectation value, v. So we rewrite this field as v+h, separating the v.e.v. from the excitation, and watch the constant v.e.v. provide a mass to the fermions, linking left and right chiral pieces thereof. The h is leftover "debris".
But you have a point, in that this Higgs particle, h, therefore couples identically to its v.e.v. to the fermions, also linking their chiralities: so, when those fermions emit higgses, virtual or not, flipping their chirality, the strength/Yukawa-coupling of such interactions is exactly proportional to the fermion mass that this v.e.v. effected!