viewtopic.php?f=134&t=30441&hilit=thermostat+fans https://www.youtube.com/watch?v=tfj3dUZTYJYRichardH if you look at these posts they might give you some background info.
Using them I think I'd do the fan and auxiliary fan sensor tests first, expecting the tests to show them working as expected, then suspect the CTS or 'fan switch'.
These are expensive to replace or use as a substitute; 'peteoddcar' on this forum has put together a simpler, cheaper alternative that's working permanently (now) on his m100 and would certainly be a cheaper way of substituting this sensor (or perhaps a used unit might be another way?)
When you read these posts you'll see that there's another sensor that's part of the picture. It's an ecu/temp sensor that's very close to this CTS unit - and under them all is the thermostat. Imo these, too, are candidates that can cause cooling 'upsets'
Keep us informed of your quest, feedback is the lifeblood of this forum. (This is how we all learn about our m100s. Quite often folks ask for help then go silent and we don't know if our advice has been relevant or useful...)
Also...'ElanScan' is a really good tool that can show you if you're getting to operating temp. If you're ticking over at ~1000rpm I suspect you are - and that might rule out a thermostat issue. Perhaps someone nearby might 'plug you in'?
edit - just a thought - is the white nylon push-on connector to your CTS 'sound' and not loose? Or tightened up okay to earth properly on the engine block?