.....and you see that that transaction is accepted on the block chain, then you KNOW that A has gotten 5 coins more in his balance
No. You do not know.
Of course you know. This is exactly the same as when you have paid cash or gold to someone. You KNOW that he got it. He knows it too. He can deny it, but he knows he's lying (or has memory problems). You can deny it too, that you paid him, but then you know that you're lying.
What you are doing here, is requiring proof to a third party. But the two involved parties know exactly how things went. If I give you a $100 bill, I know you got it, and you know I gave it to you. That's the "balance check". One of us, or both of us, can lie about it to a third party. But that's something else.
That said, as you point out yourself, in Monero you CAN prove to the world that you did pay that person, while with cash, there's no such proof available.
But this has nothing to do with the discussion about checking whether the balances are right, that is to say, whether the balances are the result of a list of transactions. Your initial statement is that *one cannot have the same credibility in the VALUE PROPOSITION of an obfuscated block chain rather than in an "open book" block chain* because one cannot "double check" the VALIDITY of the accounts by an audit. You compared this to "banking accounts" that are audited by an army of accountants.
I indicated to you that the validity is essentially the verification that the accounts are the results of VALID TRANSACTIONS, and valid transactions are such that they ensure that they are right/power to spend by the payer, such that there will not be/hasn't been double spending, and that there is a succession of valid transactions that leads all the way back to a valid creation. So essentially, a valid transaction comes down to a proof of no double spend, and legitimate origin.
The way VALID TRANSACTIONS are implemented in different monetary systems are varied but as much as these different systems are correct and trustworthy, one can rest assured of their validity.
With gold, it is the laws of nature that make for valid transactions. The origin of gold is "open" in the monetary system of gold (it doesn't matter where it comes from: dug up, from space, produced with nuclear transformation, the Stone of Wisdom...). Everybody has confidence that the laws of nature implement a correct transaction, that you cannot "double spend" gold and that holding gold gives you the right to spend it.
With cash, this is almost the same, except that the origin has to be a printer at the central bank. If the bill is sufficiently sophisticated, and if one has faith in law enforcement killing off enough counterfeiters so that there isn't a big source of false bills, then the laws of nature, and a check on the bill are also a system that allows for the implementation of a correct transaction, the same way as gold.
Bank accounts are another beast all together. The central notion is "a balance" and of course, balances do not necessarily implement correct transactions. So an army of accountants is needed to check for that. But that is because of the nature of bank accounts which are just balances, and do not follow automatically from transactions, although they should. The balance holder can cheat (your bank can display what it wants, your exchange can display what it wants), and so this must be checked by accountants in such a way that you can trust them somehow. This is in fact the weakest monetary system, but it is the one of banks. People accept it most of the time, except in the Weimar Republic, in Zimbabwe, and a few other places where the monetary system lost trust (hyperinflation).
Bitcoin goes back to transactions. The validity comes down by checking explicitly the succession of valid transactions and the validity of creation. Double spending is checked by explicitly verifying that the spending didn't happen twice in the list of transactions.
Monero does about the same, except that the check is cryptographic instead of explicit.
As to the anonymity, the whole IDEA is that transactions are confidential between the payer and the payee. Just as with gold and with cash, you cannot have both that this is public, and that this is confidential. If it is confidential, of course you cannot check it when you are a third party, without agreement of the two parties.
But, as you yourself outlined, monero has some possibilities to prove to third parties the bad faith of one of the two confidential parties, contrary to gold or cash.
However, this doesn't put into doubt the validity of the monetary implementation, which is necessary to give credibility to the value of a right to spend in the system. Otherwise, gold wouldn't have this either.
