You people are amazing, coming up with things like the rocket-sled seat-test photo. Multi thanks!
The F-104 did have a downward-firing seat briefly, and it was actually used at least once in anger, but they did change it pretty quickly to a conventional seat. Looking at diagrams of the F-107A structure, I suspect the problem would have been that the nosegear retracted forward and up, and the truck ended up right under the pilot.
Thank you for enjoying the 10-best-landing piece. Look for my slightly controversial Amelia Earhart article in the current issue of Aviation History.