Starter Engages While Driving

Actually I did install an in-line fuse. Forgot about that. Another $5 or so.

Years ago I jury-rigged an unfused electrical circuit on my truck. One day I was loading the truck with some wood and the wood shifted and shorted that circuit, and the wire caught on fire. It was touch and go there with the wire on fire and the load of wood, but I managed to break the wire before things got out of hand.

My 1965 Rambler had this problem. In the 8 years I owned the car, I had to replace the ignition switch twice to prevent the starter from cominb on after the engine was running.
I still prefer the system that was in both my 1947 Pontiac and my 1950 Chevrolet pickup truck. Both of these vehicles had a starter pedal on the floor. Depressing the pedal pushed the starter pinion into the flywheel and closed a switch. There was no relay nor a solenoid to go bad. The starter system was completely independent of the ignition system. The starter would operate without switching on the ignition.
The 1948 Pontiac with the Hydramatic automatic transmission had a rod from the starter pedal to the transmission linkage. Stepping on the pedal put the transmission in neutral. There was no neutral safety switch to go bad.