Standard Model: Problem with Masses of Elementary Particles?

In his book "Modern Particle Physics", Mark Thomson explains two problems with masses of elementary particles in the SM:

(i) If we take the QED Lagrangian $mathcal L = bar{psi}left( igamma^{mu}D_{mu} – mright)psi – frac{1}{4}F_{munu}F^{munu}$, where $D_{mu} = partial_{mu} + iqA_{mu}$ is the covariant derivative. Introducing a mass term of the form $frac{1}{2}m^2_{gamma}A_{mu}A^{mu}$ would break the required invariance under $U(1)$. This I understand so far.

(ii) On page 469, in Chapter 17.4, he writes:

The problem with particle masses is not restricted to the gauge bosons. Writing the electron spinor field as $psi = e$, the electron mass term in QED Lagrangian can be writtern in terms of the chiral particle states as $$-m_{e}bar{e}e = -m_{e}bar{e}left[ frac{1}{2}left( 1-gamma^{5}right) + frac{1}{2}left( 1+gamma^{5}right) right]e$$

Short comment from my side: On page 142, he wrote that:

[…] any spinor $u$ can be decomposed into left- and right-handed chiral components with
$$ u = frac{1}{2}left( 1+gamma^{5}right)u + frac{1}{2}left( 1-gamma^{5}right)u. $$

Okay, continuing with the calculation on p. 469:

$$-m_{e}bar{e}e = […] = -m_{e}bar{e}left[ frac{1}{2}left( 1-gamma^{5}right)e_{L} + frac{1}{2}left( 1+gamma^{5}right)e_{R} right] = -m_{e}left( bar{e}_{R}e_{L} + bar{e}_{L}e_{R} right) quad (17.16)$$
In the SU(2)$_{L}$ gauge transformation of the weak interaction, left-handed particles transform as weak isospin doublets and right-handed particles as singlets, and therefore the mass term of $(17.16)$ breaks the required gauge invariance.

Question:

How can we understand his last sentence?