Actually I think that's already here. The banking system is now modernized because you can connect your card into your Paypal account. It seems a success and there's a discussion in this forum already about Paypal Vs. Bitcoin. Both of them have their downside and advantages but I think Paypal which uses fiat for digital transactions are doing just fine. And since Paypal funds is like fiat's cryptocurrency, it's accepted in most online stores unlike bitcoins.
Paypal is still a different thing. Paypal is something global, privately owned, and still, relies on fiat currency. It is similar to credit cards and technically it still uses credit cards as its payment methods. It's just a payment processor with additional features. It's no different with you paying with credit card at your local gas station; it just uses a third party to pay for something. And technically if you use a credit card with PayPal, there are now four parties which is the payor, the payee, the bank and PayPal. Additionally, this adds more cost as you do not actually pay directly to the seller.