But for this to work you have to trust the issuer to repay the bill in fiat (or better in BTC to your own wallet).
No you dont need that kind of agreement with the issuer. You could make your own keyless bills, load them with the denominated value and pass them to friends (gifts), waiters (tips) etc. Your friends can keep them or pass them on. That way its keeps circulating - like gift coupons or like cash money.
In that regard I could photocopy a hamburger and feed the world...
With no way to do anything with the bitcoins, the paper is worth zero bitcoins.