Other_E,
I found the following notes on the RossTech site. It appears you already have their tool, thus I recommend following some of the notes on the list below.
If you have a voltage meter, you may be able to check the harness points to see if there is any drop in voltage from source to target.

Possible Solutions
Check G62 - Engine Coolant Temperature Sender
Inspect wiring for G62 (Coolant Migration, damage, corrosion, etc)

Check (if applicable) G83 - Radiator Outlet, Coolant Temperature Sender
Inspect wiring for G83 (Coolant Migration, damage, corrosion, etc)

Check Coolant Thermostat
Use Graph function in VCDS, Measuring Value Blocks - MVB
Start Engine cold, graph Coolant Temperature Sender (G62) Typically MVB Group-001
Hold RPM around 1,500rpm and watch coolant temp increase in Graph until thermostat opens, indicated by a slight drop in coolant temp
If coolant temp drops below minimum operating temperature (80°C) or never reaches operating temperature, (verified by external thermometer) suspect failed thermostat

Vehicles with both G62-Engine Outlet, Coolant Temperature Sensor and G83-Radiator Outlet, Coolant Temperature Sensor
Use Measuring Value Block group MVB-130 instead on MVB-001
G62 should increase in Temperature before G83 increase when engine coolant is cold
G83 will increase when the thermostat opens and after G62 reaches operating temperature (80°C)
If G62 and G83 both increase while engine warms up, suspect failed thermostat


Ross Tech Notes on P2181


For my 70K mile service the tech also replaced any wires he found that appeared to have cracks. Most appear to be ok, but some vacuum lines had to be replaced. Would not affect the fan going off for sure. I've had the fan go crazy occasionally and once I cleared the codes the fan would stop messing up for weeks. I am hoping it is all resolved with this big service which also included change of some of the items noted in the above list.
Good luck and keep us posted.