What would you do when the FAA shows the insurance company as the last legal owner and they refuse to sign off ?
Yep, I've had that happen on cars too. You can usually publish a 60 day notice of ownership statment, based on the vehicle/vessle/aircraft as abandoned and send a copy to the last published owner by certified mail. If they want to contest, then you get to go match up paperwork, if they don't contest, then you get the goods. When you go match up the paperwork, the putative 'owner' usually has papers showing he sold it to the next guy in line, but doesn't want to give up paper ownership. The states frown on this, as it clearly is a way to sell the asset for money, but retain ownership rights, which is a clear no-no.
If it's a case where the asset was not sold but stolen, then you loose your asset, plus you could face criminal charges depending on if you bought it in a good faith way, and paid existing market value on condition. For example, if you bought a 2005 Lamborghini in pristine condition for $10,000, it could be argued that a reasonable person would expect the car to be stolen. However, if you paid $180,000, and had a written bill of sale with the owners bonifides on it, you bought it in good faith, and all you loose is the money/asset.
In some states, cars older than 25 years go off the title system, and revert to a simple bill of sale, where no title history is checked. This is not the case with aircraft, as the FAA will always record a properly construed lien, or covenant. Which is nice, because the history is pretty much always there, and you can trace stuff. Where it gets tricky, is in cases where the asset(plane) has changed hands through estate of a deceased, or even worse, when a company/corp has gone out of business. It can still be done, but requires more legwork, that may be costly.