I'll look at the coolant sensor...Fan does run, no light, especially when stopping after a ride at 95+ degrees. '91 RS does the same, K12 must have a larger cooling system as the fan rarely comes on.
This is an old bike. The mileage tells us that it has probably been neglected. You're already aware these bikes should be ridden plenty to work well.
The overheat warning light won't come on while the fan is working unless the coolant level is critically low, the thermostat is seized closed or the whole cooling system compromised by corrosion and degraded coolant. The bulb isn't lighting probably because the fan is working as designed. It's cooling. You've got to do more than "look at that coolant temperature sensor" though. You've got to test how well it works in gradually heated water by using a multimeter. The instructions are in the K100 troubleshooting manual. You can always check the bulb to see it it lights.
This neglected machine has maintenance needs that should be rendered asap. The crankcase Z-hose should be checked. The intake manifold rubber couplings should be checked for leaks using methods available on the site. The valve clearances should be checked and adjusted if necessary. After that, if the air filter is clean, the throttle bodies should be balanced, the correct idle should be set. The coolant system should be flushed then new long-life coolant without nitrites should be mixed with the correct proportion of distilled water for your climate and added to it.
The bike sounds as though it's usable as is but taking care of these task eventually and in their proper order should make it run optimally. Like your first '85, my bike only needs the choke in first position below 60ºF and then only for as long as it take me to get into second or third gear. Then, it gets shut off and the bike is ready to shift up shortly thereafter. The choke's second position gets used at 40ºF or below. It's not a "choke" but like you pointed out, that's what BMW calls it except in one workshop manual it might have been called a starting speed adjuster.
That is just too much effort for a lot of people to remember and update their Facebook pages at the same time. "SSA"? That's a possibility. :giggles
The coldest temp I've ridden this fall so far was 28ºF.
