Coinbase and al. also do exactly this though: when you buy bitcoin on an exchange, Coinbase gives you an IOU for bitcoin, with zero transaction recorded in the blockchain it's only when you want to actually take these bitcoins out of your Coinbase wallet, to move them to your own bitcoin wallet, that a real bitcoin transaction exist. In the meantime, there can be more bitcoins owned on Coinbase than real bitcoin in existence, exactly like a bank. (And commercial banks have existed is the US long before the Fed, which was relatively late invention (1913) designed to avoid /limit the scope of financial crisis).