I believe this is called Bill Presentment. I get this on my checking account and find it quite convenient. I get an email telling me the bill has arrived and when they will pay it, and an opportunity to intervene. I don't want the vendor to be able to pull money directly from my checking.
My only problem with it, my checking account doesn't have this available for all my monthly bills, in particular some big bank credit cards.
I'm just wondering if the list of compatible vendors vary for different bank's checking accounts. I would switch from my current bank if there was one with more coverage. My guess is that they all are the same, and maybe actually farming out the service to the same service company.