I don't think it's actually related to a network problem:
I'm really as puzzled as you as to why this happens, but it seems like the app is trying to launch one of its helpers (those that live in the `bin` folder) and can't - the "connection" is between the app and the helper, and it seems like it can never quite be established.
I'm still suspecting antivirus foul play here - they tend to have apps run in sandboxes, intercept what they're trying to do, and decide on the fly whether or not to allow it. If any of your antivirus decided to disallow it, it would result in strange behavior like that.
Do you have a way to whitelist the itch app completely? ie. add its folder as a safe location? (or at least %LOCALAPPDATA%\itch and %APPDATA%\itch\bin ? so that it's only the app itself, not games installed with it)