Elitzur's theorem [Ref. Andreas Wipf, Statistical approach to quantum field theory] states that
A local gauge symmetry cannot break spontaneously. The expectation value of any gauge non-invariant local observable must vanish.
Consider a spontaneously broken ${\rm U(1)}$ gauge theory of a charged scalar field coupled to the electromagnetic field $$\mathscr{L}=(D_\mu\phi)^*(D^\mu\phi)-\mu^2\phi^*\phi-\lambda(\phi^*\phi)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\tag{1}$$ with $\lambda>0$ and $\mu^2<0$. When the field $\phi$ with the polar parametrization $$\phi=\frac{1}{\sqrt{2}}\big(v+h(x)\big)\exp{[i\zeta/v]}\tag{2}$$ plugged into Eq.$(1)$, the field $\zeta$ disappeaears from the theory upon making a suitable gauge transformation. Therefore, there is no Goldstone boson.
In this popular derivation, the gauge non-invariant field operator $\phi$ acquires a nonzero VEV in violation to Elitzur's theorem.
Question What is the reason for this apparent violation of Elitzur's theorem?