Bill pay provided by credit card companies such as Chase and Amex is a common service. Not all merchants accept a credit card though, and I'm not aware that any credit card company's bill pay will then pay the bill; that would sort of be equivalent to a cash advance.
I don't use bill pay, whether through my bank or any of my credit card companies. While having all your eggs in one basket is generally viewed as easy, the downside is the service may increase the potential for marketing pressure. I can do without it 
Instead I use the payment service of each merchant. Everything whenever possible is on auto-pilot to be paid in full. For the grand overview, I have a spreadsheet pie chart. The bank and each card (I have few) offer annual statements of charges by category.
