You may be chasing something down the wrong road here. Your tests seem to indicate that your cylander head isn't grounded. Must be a flaw there somehow. Why would you lightly thread in a spark plug? My guess is that you should put in a new spark plug, get the old fuel out of the tank,lines and carb bowl and mix fresh fuel and life will be good again.

Not clear why, you indicate chasing something down the wrong road here. What do you mean?

What I can't figure out is that since this is a 2-cycle engine the cylinder head and coil mount are part of a single unit, which should mean it must be grounded. The only reason to lightly thread the spark plug is to avoid reassembly completely. I have used to testing procedure with a new NGK plug (as I'm not a Champion fan; too many duds encountered) and fresh fuel. The results are identical as previously described. I did test with the plug torqued to the spark plug hole, but the result was the same. Upon retesting, I cleaned the coil mounting points with emery cloth. Hope you can point me to alternative tests or procedures to check.

The only reason a plug wouldn't spark when screwed into the head would be compression.
You have a marginal ign. coil or plug. There's nothing else it could be!

What I meant to say is it isn't likely a spark condition. If you have spark with your timing light, you have spark. Will it fire or try to run with a shot of carb spray into the venturi of the carb? Try it. If so, you need to give the carb a clean. Usually no starts on these are carb/ fuel related like anything else. Or run w/o enough oil and now has low compression.
Seeing tho the module was replaced, does it only install the one way? I remember that some of the early Suzys on the WPMs had an updated module where you changed the flywheel with it. That is, if my memory is correct. It's been a while. Confirm that the module you installed is the correct one. If you have further question, please post model and serial #s. Good luck.

Here is a link that might be useful: Toro parts

I believe it is a spark problem, since testing today I tested for spark, without the carb or fuel tank. The results were the same. With a new spark plug grounded, a strong spark. Then with the spark plug torqued, no spark at all. All switch wires disconnected, and contact points cleaned. Prior to testing I checked primary lead and coil grounds using an ohmeter.

The snowblower is a Toro CCR 2000E, model 38186, S/N 2000151
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Prior to the last tests, I did clean the carb using a commercial carb cleaner bath. Although as stated, the last test was without the carb installed. My next course of action was to disassemble crankcase, first removing flywheel checking for a partial sheared flywheel key. Nothing wrong with key found. If nothing else this will provide the opportunity to thoroughly decarbon.