FRB succeeds because these institutions accept each other's IOUs, withdrawal panics can be stopped with banking holidays, big banks will bail in each other to advert a crisis and then bail outs from the government who consider these banks too big to fail.
MtGox wasn't a bank but imagine if there were other MtGoxes and they were chummy with each other? They would had rescued MtGox and there may had been no Bitcoin bust in 2014.
Impressive edit, my question still stands if you're able to answer it?
Best way to answer it is for you to deposit funds into multiple exchanges and try following it on the block explorer. They use scripts to break up deposits, have their own mixers, move them into multiple cold storage wallets, et cetera - but none of this is actually mentioned on the interface at the exchange's URL.
They can also sell Bitcoin for $Fiat and other coins without necessarily proving they have those funds (beyond putting forward an address to one of their cold wallets which is no different than the local teller at your bank showing they have $100K in cash on site despite accounts numbering in the millions).