Many people want to do what you are suggesting: to have one company write checks and pay all the monthly bills. Some banks do this for free, but in most cases, you have to pay for this service.
But there is another way to get the same thing done. Virtually all monthly bills give you the option to automatically debit your account 100% free of charge. Instead of having one company pay all your bills, have all the bills charged to that one company. Do it in reverse. It works with almost all bills and is free in every case that I've ever seen.
Don't have your bank write a monthly check to your water/garbage/cell phone/internet/tv companies. Instead, have your water company automatically debit your bank. Have your garbage company automatically debit your bank. Have your cell phone company automatically debit your bank. Etc.
Or even better, if you pay your CC bills in full and on time: have the bills autmatically charge on to your CC and get the 2% cash back.

Of course, not all people pay the CC on time, so this advice isn't a good idea for everyone. You can even have that CC automatically debit your bank.