I've never seen a credit card that did not allow you to schedule your payments. If you auto-pay everything with credit cards, then by default you are scheduling your payments. Same with mortgages. For the few things that don't accept cards (rent/water/gas/electric?) you could schedule payments whenever you want through your bank, though you'd likely be paying in advance rather than in arrears.
I've never paid bills from the CC side, but from the provider side. I don't even know how I'd tell my CC to automatically pay some entity on a set schedule. It's all pulled by the service providers, separately, on whatever schedule they decide (or choose to let me configure). I know I can do it with account-to-account transfers (so, electronic checks) from my checking account, but had no idea that was a thing for credit cards. Where do you tell it to send the money? Do you have to get bank account info for the receiver?
[EDIT] On reflection, I've even built subscription systems reliant on credit card payments, and didn't know you could do this.
You don't need to tell your CC to pay some entity on a set schedule. I don't think my initial post was clear enough.
Credit cards are paid in arrears, usually with a 15 day grace period, so you are effectively getting a 15-30 day interest free loan on all purchases. When you set up a credit card, you can select any payment date you want. So, if I get a new AMEX, I can choose to have the bill due on the 1st, or 12th, or any day I want.
If I pay all my bills (i.e. Netflix, car insurance, etc.) with my AMEX, I know the AMEX bill is due every 12th (or whatever day of the month I choose). For any bill paid by the credit card (i.e. Netflix, car insurance, etc.), it doesn't matter what day of the month that particular bill gets charged to the AMEX. I only pay AMEX once a month, on the day I selected. So, by choosing what day my AMEX gets paid, I am effectively choosing which date I am required to pay all of the bills that got charged to my AMEX.
There are very few things that you can't pay by credit card. For those (i.e. mortgage, some utilities), you can schedule auto-payments on any day you want, through your bank.