Every book on the Standard Model introduces early on the concept of left and right-handed quantum fields, defined as \begin{align} (\psi_L)_{\alpha} = \left(\frac{1-\gamma_5}{2}\right)_{\alpha \beta}\psi_{\beta} \\ (\psi_R)_{\alpha} = \left(\frac{1+\gamma_5}{2}\right)_{\alpha \beta}\psi_{\beta} \end{align} From here left $SU(2)$ doublets such as \begin{pmatrix} \nu_L \\ e_L \end{pmatrix} are defined, with $\psi$ changed by the corresponding particle type letter. Right-handed fields are singlets under $SU(2)_L$, so no doublets there.
From this point onward, all GUTs (like Pati-Salam, SU(5) or SO(10)) are said to contain $SU(2)_L$ and to explain it by embedding it in some larger group and getting it back through symmetry breaking.
My question is: what does it mean that, for example, the Pati-Salam model explains $SU(2)_L$? To me, the only thing it predicts is that some fields transform as singlets and some others as doublets under $SU(2)$. The fact that the doublets contain only left-handed fields is kept entirely unexplained.
It somehow appears as if "transforming as a doublet" is taken to be the equivalent of "being left-handed" but that's not right. Left-handedness comes from spinor indexes, which are completely unrelated to the $SU(2)$ rotation taking place.
