Are you serious? Do you think "we should", really? Have you ever considered the problems and probably impossibility in achieving what you want?
Think about time spent counting, ensuring that there are no falsified notes, bringing excess notes to the bank and making sure there is change, etc.
I think it is quite feasible to create an electronic payment system that has comparable overhead.
Consider that there are existing electronic payment systems, e.g. electronic cash in europe, that already work and are quite secure (being smartcard-based)
The only way besides having centralized for-profit organizations maintaining the payments infrastructure is to use cryptocurrencies.