Why are there no particles with charge, but no isospin? The standard model has two parallel series of leptons: the charged leptons (electron, muon, and tau) with -1 electric charge and -1/2 weak isospin, and the neutrinos, with 0 electric charge and +1/2 weak isospin.
Why do we not have a third series of particles with non-zero electric charge, but no isospin? I.e., particles with purely electromagnetic interactions, and no weak interaction? Does something else horribly break if such fields are added to the standard model?
 A: Nothing would break. The theory would be renormalizable.
The reason why we have the peculiar set of particles and forces we do is unknown. One speculation is that they are a random consequence of how extra dimensions happened to compactify after the Big Bang. If there is a multiverse, other universes might have totally different sets of particles and forces.
A: One has to distinguish between left-chiral electrons and right-chiral electrons.
Left-chiral electrons have the quantum numbers as mentioned in the post, i.e. they have non-zero electric charge $Q=-1$ and weak Isospin $I=1/2$ and $I_3=-1/2$. However, right-chiral electrons actually only interact weakly (and electromagnetically) via the $Z_0$-boson, but not via the $W^{\pm}$-boson, i.e. they have non-zero charge $Q=-1$ and weak Isospin $I=0$.
This is the chiral character of the Standard Model.
Important detail: left-chiral electrons have weak Hypercharge of -1 and right-chiral electrons have weak Hypercharge of -2: $Y_W = 2(Q-I_3)$.
I speak here of electrons. For their anti-particles particular rules (more or less everything is swapped) apply.
