I have an html game and a site where that game is hosted. (example.com).
Would it be okay to upload only index.html and direct user to example.com.

As html games run in iframe here, user would not really notice it, but technically user is routed to different domain (from itch io "assets site" to game URL.)

Sure, there are some multiplayer games on itch.io working like that by necessity. There might be some security issues, but those can be fixed.