"Is it possible to design a network that doesn't expose where information is flowing, or better yet, doesn't even need to know where information is flowing (it can't leak what it doesn't know)?"

Post encrypted messages to Usenet; since anyone can receive them, there is no need for a destination address. Post the messages through anonymous remailers (mix-nets) if you want to avoid revealing that you sent them.

This problem was solved a long time ago.