I think that I know the reason why the blockchain network is not totally anonymous. People who use bitcoin are always hiding behind their bitcoin address and also their personal information, so no one can know you exactly your location or what is your name or specific details like that, but they know how much do you have in your address and they can see your transactions. I don't think it will be right if you also hidden this thing, so then you can't even make any proof that you are already sending/receiving bitcoins.
You know, even if you did use a privacy/anonymous coin, the premise of the "system" you mention doesn't change. Transaction with anonymous coins can still be proven and verified, if coins were sent or received - only that in most cases, this information is privileged to parties to the transaction. So yeah, you couldn't look up the balance unless you were given access, but the anonymity features don't remove auditability, they merely restrict it.