Why do Instructions to Leviton nylon plugs and connectors say “DO NOT use with solid wire”?
Solid Romex style electrical cable is not designed for use as portable extensions or cords. The main reason for this is that the solid wires are not flexible and will eventually crack if flexed too much. The cable jacket is also not designed for it can be moved around, stepped on or driven over. 

So with the understanding that the cable has to be flexible stranded wire it is also a requirement that the plug ends attached to the cable have to be designed for stranded wire as well.