Now Charlie gets $10 dollars as credit (not the same as cash) from Alice (mediated through Bob) and wants to buy $10 items from David and Eve. He shows the letter saying that Alice is good for the cash. Without a distributed consensus method, how can David or Eve confirm that they will be able to go to Alice to collect the cash? How can they even know that Alice has the cash in the first place? What if something happens to Bob?

The only way to do this without a blockchain would require something like Paxos, but Paxos only works if the participants are selected a priori. Every new participant would have to be vetted by everyone else, or everyone else would have to follow some central authority that can grant access to the system. If you are going this route, you are just re-inventing a credit cooperative.