You're responding to wrong comment. In the comment you meant to respond to I admitted I wasn't sure whether continuity was required to make the proof go through and you still haven't demonstrated that it is. Riemann surface here is just a fancy word for the loop itself and indeed the key part of the definition is connectedness not C.

You linked a math SE question with no responses (yes I read the comments), Arnold's book, and the same proof I've already read. So I'm not convinced.