The SU(2) you are talking about is called weak isospin, and corresponds to conserved currents in the EW lagrangian, similarly to QED and QCD. As you said, its generators flip members of isodoublets to each other.
So, for instance, its $\tau^+$ acts on a left-handed electron and yields a left-handed electron neutrino. That is, the SU(2) doublets of the theory are $(\nu_e,e)^T$. Likewise, the left-chiral quarks fall into such doublets, $( u,d)^T$, etc...
The vacuum of the SM is in a funny SSB phase, and, unlike the lagrangian, is not invariant under that group, and so the charges corresponding to the currents are not quite well defined, and largely not conserved. (You could detect their shadow conservation ghostly poltergeists, if you are very careful, but let's not go there...)
To complicate matters, there is another group, a weak hypercharge U(1) which also couples fermions, and the SSB mixes it up with the 3rd isospin component of the above, in a beautiful mesh.
The $\tau^\pm$ pieces of the currents/charges were well-understood to describe β-decay at least a decade before the advent of the SM, by Feynman and Gell-Mann, something like terms $W^+_\mu \bar\nu\gamma_\mu (1-\gamma_5)e$, etc, but the $\tau^3$ pieces arising in the commutators thereof with the hermitian conjugates appeared to specify interactions which were not there... until Glashow unravelled their peculiar symmetry mixing structure... Weinberg & Salam organized them, and finally the corresponding "neutral current interactions were observed at Gargamelle, just as predicted.