No, the mortgage holder has a lien on a property, which can be assigned. Title always remains with the lender.
If ownership of a lien is unclear, it may be unclear who ultimately owns the mortgage, but the mortgage servicer should remain. Not a lawyer, but I believe the laws relating to the owner/mortgage servicer relationship are designed to maintain consistency. A mortgage holder needs to be able to show proof of ownership. In the 2008 era, there were cases where this was lost and homeowners successfully challenged and had liens removed.
Well they could loose so much you essentially end up squatting in your own house. And if it happened to enough people you could hopefully just wait it out until you had a claim.
But America being America. You'd have a local sheriff there smash down doors with the local SWAT team to reclaim the assets for someone who bought the debt on the cheap.
in the 08 crash there where some people that won their house in court, due to the industry using a practice called Robo-Signing. Some judges saw this as irreparably separating the mortgage from the deed. So while you technically still owed for the mortgage you signed, the asset could not be reclaimed due to the fracture of the mortgage from the deed due to robo-signing. The practice was used to move mortgages around while not having to pay filing taxes every time it changed hands. It robbed local municipalities of a lot of money in avoidance. So some judges frowned on the practice and basically awarded the house to the owner. Whether you would win your case or not was hit or miss though.