You missed the point: The goods in the bank were valuable primarily because they were currency. So in fact, the only value the bank-issued currency had was social value, rather than a real value. Ergo, the paper money issued by banks had no more real value than money issued by the Federal Reserve.
Yes, it's possible to have a world where everything is privately owned and managed, including money. Here's why we don't do that:
1. I buy 100 Quatloos worth of goods from you, and pay my debt with a 100 Quatloo note from what appears to be a perfectly sound Smith Bank, which you accept.
2. Smith Bank, for reasons that have nothing to do with either of us, goes out of business, and its assets no longer exist.
3. Because everyone knows Smith Bank doesn't exist, no one will exchange your 100 Quatloo note for anything else, so your note is now a worthless piece of paper, and I just walked away with 100 Quatloos worth of your stuff without paying you anything of value.
4. You, being a smart businessman, decide to hedge against the risk I just demonstrated exists by selling me only, say, 95 Quatloos. In order to judge that risk and price correctly, you have to have an accurate picture of the financial soundness of any bank who's paper I offer, which is plainly impossible - even if you knew all the area banks, you would have to be omniscient to know what their risk of failure actually is.
5. Even if you did have that understanding of each bank's soundness or lack thereof, now I don't know what your pricing is until I actually go to pay it. Any concept of fixed pricing (which enables you to go to a store knowing how much a can of beans will cost you) goes out the window.
6. It gets even worse: If the other banks in the area served by Smith Bank want to drive Smith out of business, all they have to do is collectively refuse to accept Smith Bank's notes. Smith Bank doesn't typically have enough bullion to pay for all the notes it has in circulation plus all the deposits in its accounts, and because all the nearby banks are questioning Smith Bank, all Smith's noteholders and depositors will notice and trigger a bank run.
These objections aren't purely theoretical: All of this happened as a result of the Panic of 1837.