If you're looking for inspiration on how bills should be paid, just take a look at nearly any other country.
Money can be transferred between people in an instant, with no hassle. Bills can be sent directly to your bank account, and you can just choose to pay them, or dispute them if you like, one click, that's it. It's rare to spend more than even a minute or two on bill payments.
The systems exist, it's just the slowness or unwillingness to adopt that's the problem.