This is a very good answer which will probably convince me after thinking it over. Meanwhile there are still the two subsidiary questions: 1) what is different for spaces of curves, and 2) is there a way to formalize the objection in the above note, from other than a constructivist perspective.

The quotient is a Banach manifold (or Hilbert manifold?), away from the reducible connections (which are a major part of this question), but that does not imply paracompactness.