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.
It's significantly related to Riemann surfaces.
https://math.stackexchange.com/questions/3349111/where-exact...
https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.p...
https://projecteuclid.org/journals/topological-methods-in-no...