A lot going on here so....
First, when pushing in the key switch (choke) you should here the solenoid "click". You are correct in the fact you should show voltage at the wires when doing the same. Rotating the red lever by-passes the solenoid and allows fuel to be pulled into the carbs. When using that method, returning the lever to the closed position would be best after warm-up. You seem to have that down pat and it would appear that there is a "disconnect" between the ignition choke portion of the switch and the primer solenoid. Could be a switch problem (dirty contacts) or a break in the wiring to the primer.
You might try a squirt of contact cleaner shot through the key entry or checking the connectors on the rear of the switch. Hopefully, at worst, you'll need to replace the switch itself and not have to chase down a wiring connection.
Good luck.
PS.... looking at an old 1996 service manual for my 200hp shows an ohm check 5.5 ohms +/- 1.5 ohm so you could also check the solenoid coil winding itself.
