Wow. Just...wow. As a pretty savvy guy when it comes to computers (I'm a programmer for crying out loud), I'm astonished by how embarrassingly dumb this was on my part.
Everything was already in the public folder. What I forgot is that some time ago I deactivated the sharing of that folder on my client. Enabling it again had no immediate benefit though, even after clearing the browsers' caches (which I guess is understandable, since everything was uploaded through a browser).
Turns out some people were having the same problem regarding other files in the public folder (unrelated to C2), and one answer mentioned the https protocol, which turned out to be the key to this whole thing: everything works if, instead of opening index.html from the public folder (https) I open it through the "public link" (http).
I am really sorry for the unnecessary trouble. Thank you all for your answers, which proved instrumental in solving the problem :)
Have a great day!