Pretty sure most cases where people's accounts get hacked have nothing to do with the security of the website, except inasmuch as the site fails to provide an authenticator. Most "hacking" involves getting a computer virus or entering your password into a site that is not actually Facebook.
Facebook currently (or did) reset the password by having you identify the names of your friends using their pictures. I could see this as a way to authenticate a transaction.
Also not sure how secure that would be , how difficult would it be for a would be attacker to find out who was in the images?
I imagine a reverse image search could be fruitful.
The question is not if online payments is able to work, it is a question of if Facebook is able to crack the nut and reduce the fraud by using real identities. There are real problems as you have pointed out, but (and it is still a very big BUT) IF (and that is a very big IF) they are able to pull it off it will be a game changer for them.
How many of your friends have had their facebook accounts stolen? For me, it's a few percent. "Oh look, another random acquaintance is stuck in Abu Dhabi and needs me to wire them some money"
Facebook security isn't nearly good enough to be handed my wallet. Actually it's not my wallet that's the problem, it's the wallet of everybody who uses "password" as a password.