Did you bench test the starter before installing it? If not, here's a quick test you can do without completely removing it. Use a big screwdriver or a pry bar, and just push the button on the starter switch- - - - -not the entire foot pedal linkage. If the starter itself is good, the motor will spin, but the drive gear will not engage the flywheel and spin the engine over. If that's the case, you've got a linkage problem, not an electrical one. I've seen people put the twist cap off a Coke bottle over the starter switch button as a temporary fix when the starter wouldn't turn due to sloppy linkage.
If the starter motor won't spin by pushing only the switch button, start looking for open circuits inside the starter, or possibly bad contacts in the switch itself. Places like NAPA and Rock Auto sell just the switch, not the whole starter, but a "rebuilt" unit should have come with a brand new, functional switch. Good luck!
Edit: Your picture shows a black label on the starter, which is correct for a 6V unit (12V has a red label) but the switch is for a 1955 or later model 12 volt starter. It will work OK, but the small red wire in the photo is not necessary for a 6V installation. It's there to bypass a ballast resistor for a 12V ignition system, and your 6V ignition does not use a ballast resistor. Wherever that wire is going, it doesn't need to be there on a 6V vehicle.
Jerry
