All right, I was able to fix the issue by clearing all cached data for itch.io. (There was about 787 MB in the cache for me.) Since the quota was reached, there was no space left for Midnight Idle, and that was making the save-game code fail almost immediately.
If anyone else is having the same issue, you can clear your itch.io cache by going to: Settings > Privacy & Security > Cookies and Site Data > Manage Data.
Hope that helps!