There are two different types of auto payments with bank accounts. I call these push vs. pull transactions
1. Push: Electronic payment (automatic or otherwise): YOU determine who, when, and how much to pay. These are like e-checks. You can turn this off at any time by logging onto your bank and stopping the next transaction.
2. Pull: Auto-Debit: You give permission for another party to debit your account on their schedule, and assume they are doing what you've agreed with them. In order to stop this, you have to contact the business and have them agree to stop debiting your account.
The difference is that if you no longer want to do business with a company, it can be hard to stop the auto-debits. If there is an error or a disagreement, you are out the money until the problem is resolved. When I have a disagreement about my money, I want the money in MY pocket until the problem is resolved. And when it's in my pocket, companies are much more willing to discuss the issue.
I have been using Automatic payments out of my checking account for years, but I hate giving anyone else access to my account. So when companies are now requiring this, I started using a separate checking account that I only use for auto-debit and a few other things. If there is any problem, dispute, etc with that account, I will have no problem shutting it down and opening an account elsewhere. But my main checking account would be a total pain to do that with.