So I called them again. And this time, we established definitively that they never held this loan and cannot issue a lien release against it. That would be ok if they hadn't caused me to waste a week by telling me that they could do so when I initially contacted them.
I spent some time berating them and suggesting that, just maybe, it would be useful if they would actually train their staff. They won't, of course.
I have now located the former lien holder who I have had to perjure myself with on their web site by saying that I have "lost" the lien release -- I more strongly suspect that they never sent it, because it would have been in the same pile of documents where the title to my Ford was recently found -- and pay them to disgorge.
I'm less than pleased by the whole process.
On the other hand, maybe I can finally get this car into samwinolj and Bonnie's hands next week.