Are you running Linux? On Linux (and possibly on other systems; I'm not able to test such systems myself), certain components of web games (most notably the "prompt()" function) don't function correctly when run from the itch.io client. It seems to be due to an, I would argue, poor engineering decision by the team who made the backend Itch uses; see https://github.com/electron/electron/issues/472 for more info.
...Frustratingly, the "prompt()" function just so happens to be a /very/ common tool among HTML game makers to prompt the user for a custom name, so... this lack of support breaks quite a few furry games. I use the same workaround you found; if the workaround bugs you too much, see if the game functions as intended in a standard web browser.