Better to just save to a cookie.
Anyway the Idea is good and it is really fun to solve.
Without speaking for Regex Crossword (I'm in no way affiliated with them):
I suspect the reason may at least in part be security. Passwords are dangerous to keep around, hard to handle, computationally expensive if done right, and if an attacker gets at the database, people get angry. A social network provider (especially the options here, Facebook, Google, Twitter, GitHub and Windows Live) is much more likely to get security (against third party attackers) a lot more right.
One way would be to auto-generate accounts and display some random identifier to keep. Or maybe just offer e-mail to which they would send you a login link... There are other options but login with OAuth is not the worst.
No, that's a lie. No need for identification in order to save progress, you can use: cookie, browser storage, etc.. for that.
Please be honest about why you are using social media for login. Because when I login with Facebook for example, you can see a large portion of my Facebook profile and depending on the plan you have way more than that.