Higgs mechanism and neutral fields

Consider a Lagrangian $L(\phi,A_{\mu})$ with $\phi$ being some scalar field and $A_{\mu}$ some dynamical U(1) gauge field that minimally couples to $\phi$. Under a global U(1) symmetry the field $\phi$ transforms as

$$\delta\phi=i\epsilon q \phi.$$ The field $\phi$ is said to be charged (with charge q) under the gauge field $A$.

In a Higgs phase we have that $|\phi(x,t)|\neq 0$. In particular we can fix a gauge so that $|\phi(x,t)|=\Phi(x,t)$ is real. Then we consider small fluctuations $\Phi(x,t)=\Phi_{0}+\delta\Phi$ and integrate them out to obtain an effective theory in which the gauge field A is massive.

My question: It seems to me as though the requirement that $\phi$ is charged enters when integrating out the small flunctuations, because if $\phi$ were neutral (i.e. q=0) there wouldn't be any flucutations that one can integrate out and hence one wouldn't obtain a massive term for the gauge field in the Lagrangian. Is this correct? If not where does the requirement for $\phi$ to be charged enter in the argument? And: Does the requirement for the matter field to be charged with respect to the corresponding gauge field carry over without difficulties to the non-abelian case?

I am looking forward to your responses!