The OEM Manuals can be had on-line for free to $20, but I believe and/or recall they are limited to troubleshooting connections to this degree needed. But remember, DC power will ALWAYS try to find the fastest way to ground, so I'd guess you may have a poor, loose or corroded connection or a bad/worn switch internally.
And yes, I have seen plenty of key switches go bad in my life, from over-use and then from people who hang a TON of keys (weight) off the ignition switch key ring, especially when in RUN mode. This wears them out internally.
First - One of the first things I do it check ALL connections and terminations, even from back at the motor. Might also want to check operation of the start-in-neutral circuit, even throw some lube onto the switch activator if you can easily get to it. Now 90% of my experience is with OMC motors and their older pre-Ficht models used a large red amphenol connection to connect the hardness from the OB to the harness from the helm/key switch. But I once took apart such a harness connector and had to clean up the contacts and regrease with dielectric grease. Then all was well! I was getting a phantom overheat alarms, even with the OB not started, but key switch to RUN.
Ask - Do you have the older 10-wire switch or the newer 6-wire one? The Choke (primer enrichment) connector is typically adjacent to the Starter connector, so check them out. Confirm this is your switch layout BEFORE doing any tests ...
Try this, battery switch OFF (or disconnected), key switch to RUN, DVM leads to Choke & Starter connectors on switch, DVM set to continuity ... push IN on key while turning switch to crank ... and I don't think you should hear the continuity 'tone' that they are connected in a circuit or talking to each other.
Other - In the mean time try this! On my OMC motors, I prime the bulb hard, turn key switch to ON, press IN on the choke and slowly count to 4 or 8 ... as your motor will tell you which she prefers. Then release the key from IN and turn to CRANK. I do not flood the carbs when cranking, ever! And they start right up on the 1st turn of the key!
Let me know ... as I have a mid-90s vintage Merc harness/switch I can send you to try out.