The standard text worth reading is Georgi's Weak Interactions text, which outlines the effective σ models resulting out of chiral symmetry breaking in QCD. This process is quite subtle and elusive and Schwartz's book is no more schematic than standard texts. But, once you've bought the conceit of quark bilinears supplanted/summarized by mesons (and this is pure low energy QCD, nothing to do with the weak interactions; it is a potential separate question I would not be keen to answer!), which you complain about in your comment, it is not hard to see how the pions of the σ-model, this effective low energy theory, also overlap with generators of the EW symmetry, which are thus also SSBroken, in turn, but only by a little.
This, in fact, is the building principle of all technicolor theories: using states resulting out of a technistrong theory chiral symmetry breaking to substitute for the Higgs field in the Higgs mechanism and break EW. So, let me flesh out the point that @romanovzky made schematically.
Take the SM, keep only the lightest quark generation, for simplicity, so the u,d quarks only, and discard the Higgs field. Hence, u and d are now massless. QCD, in the χSB elided here (cf. WP link provided) generates a σ model that summarizes this symmetry breaking into PCAC; actually, here, CAC, conserved axial current, as the pion is massless, that is, the 3 conserved axial currents
$$
\vec{A}_\mu= f_\pi \partial_\mu \vec{\pi} +... ,
$$
where $f_\pi$ is the pion decay constant, of the order of o.1 GeV --this is low energy QCD, after all, and we are interested in features of the peculiar asymmetric vacuum.
Enter the Weak interactions. You couple these axial currents to the axial half of the W, to get schematically and cavalierly,
$$
...+g \vec{A}_\mu \cdot \vec{W}_\mu +...
$$
etc... so you are gauging the σ model, in our case, for convenience, the nonlinear one. The EW $SU(2)_L$ overlaps the 3 broken chiral charges of the Axials, so it's broken. (Recall the V-A action on pions, $\delta _{\vec{\theta}_L} \vec{\pi} \sim \vec{\theta}_L\times \vec{\pi} - f_\pi \vec{\theta}_L +...$. You can see the axial variation of the above current-W term may only be cancelled by the variation of the W bilinear in the next paragraph.)
Now the seagull term in the gauge-covariant kinetic term for the pion is, quite schematically, in the leading, pionless term, something like
$$
g^2 f_\pi^2 \vec{W}_\mu \cdot \vec{W}^\mu ~,
$$
that is, $f_\pi$ has supplanted the standard Higgs EW v.e.v. $v\sim 246$ GeV of the real world; that is, the new notional Ws now have a mass
$$
\frac{f_\pi}{246~GeV} M_W\sim 4\cdot 10^{-4} M_W \sim 32 ~MeV,
$$
real light... lighter than the strong χSB scale.
You can see how this sort of thing (which happens at some level in real life) is a negligible piece of noise in the big picture of the SM. We have omitted the "second job of the Higgs", namely giving the fermions (e.g. leptons, in case we didn't wish to descend into current quark mass complications) mass in a gauge invariant way, but it can be arranged.
Sophisticated versions of this mechanism undergird Technicolor models, where the interaction is some QCD-inspired strong coupled theory, e.g., Susskind 1979. It's eqn (15) provides the W mass "directly" through the vacuum polarization contribution of the quarks, assuming χSB, but it might come across as more cryptic than the effective Lagrangean sketch outline here.
