This system doesn’t need to be implemented as a Cryptocurrency. Just a normal database would achieve the same effect without needing to pay miners.

Another problem here is that the admin would need to store the cash locally because if he were to store it in the bank he may or may not get back the same bill unless a safe deposit box is used.

Even if all the bills are recorded online there is a high risk of robbery. People are smart, they’ll work around any limitation if there were money to be made.

Lastly, this is not a trustless system. People have to have trust in you and your team. What’s stopping you and your team from engaging in fractional reserve banking after you’ve recorded the bills? What happens if your company go out of business? From whom do user recover their money if you and your team were to disappear?