OK so the lender buys the house for its payoff.
What happens next? The bank or mortgage company doesn't really want the house. They can't live in it, and very, very few would ever rent it out.
They might list it with a realtor, or even have an in-house realtor, but in order to get back the money they lent on the house, they have to sell it sooner or later. And, if the market has declined they are going to get back less than they lent.
That is correct, but they cannot write off a perceived loss, nor can they collect insurance until there is a real loss. The job is not done until you finish the paper work. After they try to sell it at the price to not lose money and have the paper work to prove a good faith effort. Then they will sell at a reduced price if they haven't sold it to a customer by then. Most often a person who works with the lender on a regular bases buy the property.