Just a couple observations reading the thread.
In post 4, you indicate your 3rd check showed voltage on both small wires of the solenoid. You should only have voltage on the purple wire. The two black wires on the other side are grounded. The voltage between the black wires and ground should be 0. You may have an open ground. (one of those black wires makes its way to the ignition switch G terminal and the other goes to ground)
In post 7, you stated that if you unhook the ignition switch and jump the solenoid, it will start and run. I'm not sure how it could run hardly at all because disconnecting the ignition switch removes the power source to the fuel shutoff solenoid. When the solenoid is not powered, it shuts off fuel delivery in the carb.
As an FYI, when the Ign Sw is Off, the M terminal with the white wire attached is connected to the G terminal with the black wires attached. The black wires are grounded all the time. The white wire goes to the two coils. When the Ign Sw is Off, the coils are grounded, and hence the motor shuts off. When the Ign Sw is in any other position (run w/lights, run w/o lights, start), the G terminal is NOT connected to the M terminal. This allows the coils to fire. So, knowing this you can test this portion of the Ign Sw by disconnecting it and measuring the resistance between the M and G terminals with the key in all 4 positions. Starting with the key off, you should see 0, inf, inf, inf as you move the key. (inf is infinite or open circuit).
You can continue checking the Ign Sw and verify the start and run operation of it. Battery voltage comes into the B terminal via a red wire. It comes out of the A1 terminal to the yellow wires. If you check the resistance between the B and the A1, you should read inf, 0,0,0 starting in the off position. Inside the Ign Sw, the B terminal also connects to the S terminal that the purple wire is connected to. Checking resistance between B and S, you should read inf,inf,inf,0 starting in the off position. Summarizing, battery voltage is supplied to the A1 terminal(yellow wires) during run and start modes. Battery voltage is supplied to the S terminal(purple wire) only during start mode.
If your solenoid is good, you should be able to jump from the battery to the small terminal where the purple wire connects to the solenoid and the engine should turn over. Assuming the Ign Sw is good, the engine will not start (but will turn over) with the key in the off position, and it should start with the key in either run position. (PTO off, Brake set of course). If the solenoid does not kick in when you do this, either it is bad OR, you have a bad ground (or both!).
Let us know what you find!