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Well, the payment system facet of the banking sector is so tightly regulated by laws and central bank regulations that you could as well consider it controlled by a single entity - the government. Nothing much changes, and in fact you can provide by law some basic rights of access to the financial system. Currently, in most countries it is conceptually possible for an individual to be rejected by all commercial banks without compelling legal reasons.

The payments are already controlled by the central bank. How do you think a bank moves money to another?

If the financial system is a tree, the root node is the central bank, the second level nodes are the private banks and in the next level are the business and households.

By the way, this perspective makes clear that banks are not "only another business" as we hear sometimes.

This is wrong. You can be charged with crimes that bar you from the banking system. Not to mention the credit scoring system can prevent the mobility you speak of.