condensed matter - Mermin Wagner theorem in superconductors, massive Goldstone mode

For a continuous symmetry breaking one receives a massless Goldstone mode which leads to divergent phase fluctuations in 2 and 1 dimension, thus we end up with a disordered state.

However in superconductors this Goldstone mode (phase mode) can be gauged away (Higgs effect). So why is Mermin Wagner theorem invoked for superconductors? There should be no gapless mode to disorder it? Or is it some subtlety with the Higgs effect in 2 dimensions?

(A small additional question: Anderson showed that this phase mode in fact gets promoted to the plasma frequency when including Coulomb-interaction. Is this equivalent with the usual approach of "gauging away" the phase?)