Many banks, such as SunTrust, let you do "bill pay" with a checking account.
Bill pay is a part of the website which lets you pay your bills via your bank's website. You input and save the account numbers for various bills and it's so easy to pay your bills from there.
So far, I don't know of any credit cards that let you do that.
It has always been a "dream" of mine to pay all my bills to a credit card, average out what these bills cost per month, and make a big "lump" payment to the card based on the average. Like this it shows activity and builds history on my cards. Also, I might be short or over one month, since it's based on an average, but whatever is left over will just stay on the credit card and will be covered as long as I make a payment the next month.
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.
HSBC discontinued the servive a few months ago.
I prefer to have the monthly bills get auto billed to credit card, I think that is better than bill pay.
I have only 1 monthly utility bill that I have to pay via checking account bill pay. I have about 8 monthly bills that auto bill to a credit card.