I have a theory as to what is happening with the lost save data:
So I did some testing and it seems like save data is correctly saved and accessed while the browser is open, but as soon as the browser is closed (or no longer running in background like when powered off or restarted) the save data is no longer accessed correctly and prompts you to start a new game. I found out that after closing the browser, the cookies are still there but opening the game again prompts you to start a new game. Maybe the first time the game is run on the browser it is trying to tell you to start a new game? I don't know much about programming on itch.io but this is my only explanation I can think of. Like the cookies are still there it just isn't accessing the saved game.