I thought that maybe the HTTPS connection might be the problem, but it doesn't even seem to need a handshake or something similar.
Yes, HTTPS requires that you perform a rather complicated handshake, which is why essentially nobody attempts to implement HTTPS directly on sockets, and instead uses WinHTTP or WinINET, the two HTTP stacks included in Windows.
While Fiddler is running, you can kinda "cheat" by sending your plaintext request (containing a HTTPS url) to Fiddler, which will then perform the proper secure handshake with the remote server on your behalf and then return the response to your client. (Fiddler is acting as a HTTP-to-HTTPS gateway in this scenario.