It doesn't need to be the public sector directly. What I mean is that banks are not using their own money to do what they do, so it does't make sense to let them charge interest. There could just be a set of regulated institutions that charge fixed fees (independent of the amount of the operation) to perform these tasks, which basically consist of converting one form of wealth (real state or business assets) into monthly obligations.

More like reuse an existing bureaucracy instead of creating a parallel, private one.