Great question and in circumstances there are banks which are renting out the properties in order to generate income to offset their financial loss on the inventory of properties (known as OREO), but this is not the norm.
Based on the bank executives we work with at PahRoo Appraisal & Consultancy, they do not want additional legal encumbrances that come along with tenancy. Specifically, you are a renter and have rights for habitation amongst other things. Generally speaking, when you have a right to occupy the property, the banks forgoes some of their rights in exchange for your rental payment (as well as other agreements which may be part of your rental contract). Ask any landlord and they will tell you that owing a property is easy, dealing with property management can be challenging.
By not renting the property, the bank is in the strongest and most direct position to negotiate, transfer, divest or dispose of the real estate asset. As noted by fellow professionals, banks are on the in the OREO business (and hence not in the property management and rental business), but they are in the lending business. They would rather let you buy this specific property and give you a mortgage on that property than rent the property to you.
PahRoo Appraisal & Consultancy