If you're aiming at a broad consumer audience, the "Login with Facebook" button is by far the lowest-friction way to authenticate users (for better or worse).

So, the answer to your question is: "this world."

With no alternative, and nothing but a splash screen? Log in with X is certainly low-friction, but without alternatives it's still a massive barrier to many of their target users, especially when you consider this is a) Disney, and b) Facebook doesn't allow people under 13.

Is "Warimals" aimed at the under-13 crowd?

I suspect that the Facebook market penetration is high enough in their target demographic as to make this the most effective way for them to go. Sure, they could provide some secondary log-on technique for the non-Facebook-using minority, but would it really be worth the additional customer support headaches?

Don't know if it's 13- crowd - it's Disney, so at least partially, yes. I'm also assuming it's heavily Pokemon influenced, in which case absolutely, regardless of any stated aims.

As to customer support headaches: they're running an entire custom-built web game. Login problems would be the least of their difficulties, support or technical.

Facebook has something like 500 million active accounts, right? If we assume that the overwhelming majority of those have a 1:1 mapping with real-world people, then somewhere between 15% and 25% of Internet users have a Facebook account.

So, the lowest-friction way to authenticate users still cuts out 75-85% of people on the Internet.

Edit: Not to mention everyone under thirteen.