On the other hand, it's not like handling physical cash is free, really.

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)