Nope. Cookies are on.
And some games DO work. But not all. And not yours.
Looking at your comment history, you have that issue with multiple different Twine games. The issue is on your end, not theirs. Downloading the game works because it's no longer a file hosted on the internet trying to access your browser information remotely, it's just a file stored on your computer.
Idrelle Games made a post addressing some common errors; yours is number one: https://idrellegames.itch.io/wayfarer/devlog/316300/troubleshooting-game-launch-...