No, the photon is not "**the** non-chiral piece" of $SU(2) \times U(1)_Y$ before symmetry breaking. The photon is the $SU(2) \times U(1)_Y$ gauge boson that is invariant under $Q_{\rm elmg}$, the electric charge, which is given by
$$ Q_{\rm elmg} = \frac{Y}{2} + T_3 $$
where the first term is the hypercharge, the generator of $U(1)_Y$, and the second term is the $z$-component among the three generators of the $SU(2)$ factor of the gauge group. The photon is the gauge boson which is preserved by $Q_{\rm elmg}$ i.e. carries $Q_{\rm elmg}=0$, much like the Higgs condensate which also carries $Q_{\rm elmg}=0$; that's what makes the combination $Q_{\rm elmg}$ special (this is what makes the Higgs condensate electrically neutral and the corresponding gauge field, the electromagnetic field, remains a long-range force mediated by a massless particle) which shows that the particular combination is only picked as preferred by the symmetry breaking.

Obviously, one would get equivalent physics if $T_3$ were replaced by the component in any other direction of the 3-dimensional isospin space but the Higgs condensate would have to have the same new direction as well. Electromagnetism inevitably involves non-chiral physics because the chiral couplings only appear at and above the electroweak scale (the Higgs vev). If you assume that the lightness of the Higgs is explained naturally, this rule may be understood as well: all the masses above the electroweak scale are cancelled by some chiral physics when they contribute to the Higgs mass via loops; the lower-mass contributions don't have to be canceled so they may be non-chiral.

The parity-symmetric nature of $U(1)_{\rm elmg}$ (and all conjugate groups etc.) may also be seen from the grand unified starting point. One may see that the 2-spinors in GUT (or Pati-Salam) representations come in pairs with the same electric charge. What you seem to be missing is that the chirality or non-chirality of the gauge interactions is mostly a property of the fermions in the cubic vertex, not just the gauge boson. Interactions and dynamics of gauge bosons themselves is always non-chiral in 4 dimensions; only the fermions may bring the chirality.

Concerning the photino, there are several misconceptions underlying your description. First of all, photino is a neutral particle so it is more appropriate to think about it as a real Majorana fermion, not a chiral Weyl fermion. Second, and this is related, such Majorana fermions generically have masses that don't allow them to be combined into Dirac fermions with a uniform mass: the eigenvalues of the two Majorana pieces in the would-be Dirac fermion almost always differ.

Third, and this is related to the second point, it is unnatural to talk about photino because the kinetic and mass terms are generally mixed with other electrically neutral spin-1/2 superpartners, namely the "zino" and the (two) neutral Higgsinos. One must look at these four Majorana spinors, there are general mass terms and one may derive 4 different Majorana neutralinos out of them with 4 different mass eigenvalues.

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