That's just foolish - you don't need passwords of any kind. In fact you don't need accounts. You can just store cookies, accessible from this browser only, which tells you what have you solved.
That doesn't solve login in from a different device. And if you cleanup cookies for whatever reason, you lost the progress. So no, cookies don't solve the same problem as a user account (with login from a "trusted" provider) solves for saving the state of progress by a user on this game.
That's a game, an exercise for several nights
Maybe allow both options for people who want to keep their saves across multiple devices and one for users that use the same browser. It is a pretty trivial thing to do.
They already do.
Which disappears cross device or cross browser, making users flood your inbox with support emails asking where their progress went.
The website already uses cookies to store what you have solved. The cookie has 1 year expiry. Account creation is optional.