Higgs Yukawa coupling explicitly violating chiral symmetry I'm pretty confident that I'm misunderstanding something here.
The Yukawa coupling that couples the Higgs field to fermions can be written something like:
$$
 \lambda \bar L H e_r + h.c.,
$$
where $L,H$ are SU(2) doublets  of opposite charge and $e_r$ is uncharged under SU(2) and carries weak hypercharge opposite to $\bar L$ under U(1). So regarding gauge symmetries, everything checks out.
It is my understanding that the SM should have a chiral symmetry as well, so that:
$$
 \psi_L\rightarrow e^{i\gamma},\psi_R\rightarrow e^{i\eta}
$$
leaves the Lagrangian invariant. This doesn't seem to hold for the Yukawa term (after giving higgs vacuum expectation $v$):
$$
 \lambda\bar L H e_r \supset \lambda e_L v e_R
$$
But I still read electroweak theory as being referred to as a chiral theory, and chiral symmetry breaking not being discussed until QCD and Yoichiro Nambu's contribution there.
What's off?
 A: Higgs Yukawa coupling does preserve chiral symmetry (it's usually called axial symmetry).
What is missing in some text books is that  under axial rotation
$$
\psi_L\rightarrow e^{-i\theta},\psi_R\rightarrow e^{i\theta}
$$
the Higgs field transforms as
$$
H\rightarrow e^{-2i\theta},
$$
hence
the Yukawa coupling
$$
y_e \bar L H e_r + h.c.
$$
is invariant under axial rotation.
Upon spontaneous electroweak symmetry breaking, the Higgs field acquires a non-zero VEV $\upsilon$, which DOES break the axial symmetry. In other words the spontaneous electroweak symmetry breaking breaks both the electroweak gauge symmetry and the global axial symmetry.
Moreover, one can entertain chiral masses for fermions if the Higgs VEV is chiral $e^{\theta i\gamma_5}\upsilon$ (see here):
$$
m\bar{\psi} e^{\theta i\gamma_5} \psi = m\cos\theta \bar{\psi} \psi + m\sin\theta \bar{\psi} i\gamma_5\psi.
$$
However, there is another twist to the story. The ABJ chiral anomaly actually breaks the axial symmetry regardless of the existence of the spontaneous electroweak symmetry breaking. The would-be Nambu-Goldstone massless boson corresponding to the broken axial symmetry becomes pseudo-Nambu-Goldstone boson with non-zero mass due to the chiral anomaly quantum contribution. That is why the axial symmetry properties of the Higgs field and the Yukawa coupling are not usually discussed in some text books.

Added note:
The previous discussion is fine and dandy when we consider Yukawa coupling for isospin down-type particles only, such as electron $e$ or down quark $d$. When it comes to Yukawa coupling for isospin up-type particles like up quark $u$ or neutrino $\nu$, things get a bit hairy: the Higgs fields per se can not be directly coupled to right-handed $u_r$ quark, given the mismatch of hypercharge. In stead, the Yukawa coupling for $u_r$ quark takes the awkward form:
$$
y_u \bar Q_l (i\tau_2 H^\star) u_r + h.c.
$$
which breaks the axial symmetry (the complex conjugation of $H^\star$ is the culprit), as opposed to the normal $d_r$ quark Yukawa coupling
$$
y_d \bar Q_l H d_r + h.c.
$$
which preserves the axial symmetry.
On way of savaging the situation is to propose that there are two different Higgs doublets coupled to up quarks $u_r$ and down quarks $d_r$, respectively. And for that matter, this stone can kill another bird: the strong $CP$ problem in QCD. See Peccei–Quinn symmetry for more details.
