After seeing the plug spark during testing, I would have thought the wire insulation issue was not the real problem. However, I removed the flywheel shroud and wrapped a bunch of tape around the ignition wire, and sure enough, the engine fired up. I mowed my front yard with the mower without any trouble. Thanks for the help. Free fixes are the best kind of fixes.
The design on these engines makes no sense. Why on Earth would anyone ever make the carburetor out of plastic and the fuel tank out of metal? Why put the carburetor on top of the engine, necessitating a cranky little diaphragm fuel pump when it is so easy to put the fuel tank on top and let gravity feed fuel into a normal float bowl carburetor.
Whoever put this mower together gave absolutely no care to assembling it properly. There was a little spot where the metal was formed to allow the ignition wire to pass through; the wire crossed past the shroud at a different part, allowing the wire insulation to be destroyed by the vibration of the engine.