Why does the weak force distinguish left and right handedness?

I think you are sort of reversing the logic of chirality and helicity in the massless limit. Chirality defines which representation of the lorentz group your Weyl spinors transform in. It doesn't 'become' helicity, helicity 'becomes' chirality in the massless limit. That is, chirality is what it is, and it defines a representation of a group and that can't change. This other thing we have defined called helicity just happens to be the same thing in a particular limit.

Now once you take the massless limit the Weyl fermions are traveling at the speed of light you can no longer boost to a frame that switches the helicity. I think its best to think of a fermion mass term as an interaction in this case and remember that the massive term of a Dirac fermion is a bunch of left and right- handed Weyl guys bumping up into one another along the way. Conversely if you want to talk about a full massive Dirac fermion that travels less than c and you can boost to change the helicity, but that full Dirac fermion isn't the thing carrying weak charge, only a 'piece' of it is.

See this blog post on helicity and chirality.

As far as the left-right symmetry being broken people have certainly built models along these lines but I don't think they have worked out.

Does this answer your question?

The explanation is simple--- all particles we can see are chiral, they have only one handedness, because if they had both handedness, they could get a mass, and generically, that mass would be of order of magnitude the Planck mass. We live at energy scales which are teeny compared to the Planck mass, so we can only see massless stuff, so we only see chiral fermions (and gauge bosons).

The right question then is the other way around, if everything is chiral, why then do the electromagnetic and strong interactions not violate parity?

This is because the Higgs mechanism partners up the chiralities into massive Dirac particles at lower energies, and only the W,Z bosons know that they were chiral to begin with. At low energies, you get parity and charge-conjugation as accidental symmetries, because these are symmetries of the low-energy Dirac Lagrangian coupled to the remaining photon and gluons.

As for the neutrinos, a chiral neutrino can have a Majorana mass while only having one chirality, and this is certainly what is going on in nature, since this scheme predicts the mass correctly, and this mass is absurdly small.