You say, "Real, digital bank notes" but really, they are just numbers that you can write down if you want. That number is worthless unless you do the digital transfer (ie telling bank/service to do something). Realistically, it is no different. I can do all of this stuff with a bank, and person to person transfers are easy where I'm at. You are still interacting with a company or government to move your money around - and doing the same if you are receiving money. It doesn't really matter that it isn't your bank, even if you hate your bank. You have to do this even if you pay cash for a piece of used furniture: Most employers do not pay in cash.

The only real benefit to this system is that it would probably be run by a government agency and as such, would theoretically make it available to everyone. This would reduce the cost of being unbanked in the US and probably improve lives. Honestly, though, you could just make basic banking available for everyone so they can get paid, pay bills online, and shop easily - it'd likely prove to be more cost effective.