It seems like the answer then is not to have more private banks but to have postal banking and making it easier for people to have access to bank accounts. They could completely get rid of paper banking. If you pay all your bills electronically via debit cards, the bank and the person/company you are making payments to would know you have the money and reduce the risk of overdrafts.

Right now, the second you use your debit card. It shows up on your account.