So I did a bit of digging, and I'm pretty sure what happened is that the Itch.io app only allows for web browser games to be ran as a download in the WebGL application, which, since the file has to be downloaded first and ran in a sandboxed form of the WebGL window and since it seems this game was never meant to be downloaded, might be a big part of the issue here TwT
There is a way to fix it within Unity, though, but it probably isn't too necessary overall to do...
That said, it does work on an actual web browser (not Itch.io's app browser), at least- are there any plans for releasing the game as a standalone executable?