If you check around, you'll find rentals that the GSEs own.
Property managers with those rentals are normally in urban areas, not the suburbs.
When searching for rentals, start by going to the high-volume places, such as searching for Dallas Home Rentals for example, then use one of the websites you find. Smaller places may not have rental listings where you're looking.
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
The bank's sole objective is to sell the property as quickly as possible while still getting a fair price on their asset. They feel this is best accomplished by selling a vacant home rather than a tenant occupied home.
Hope this helps.
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The Michael Group - Dallas Business Journal Top Ranked Realtors
They are in the business of lending and borrowing money.
There are probably incentives, both positive and negative.
Even when the banks foreclose on a property, they generally turn it over to a management company to dispose of it.