I had this issue - resistor bad on the radiator fan. Would overheat, and then the highspeed fan would come on and keep running after it was turned off. Replacing the whole fan (resistor and all) fixed the initial cause. But it was a bear to get the coolant filled up in a non-S. Had to open the cap above the radiator. Run it until it was hot. Filled up what I could. Turned it off, did it again and again. Putting it in the overflow was not useful at all. After 3 times, it was full and fine.

 

also can try a 180 degree thermostat vs. the 195 OEM, did wonders for my 2006 r53 in addition to the low speed fan resistor bypass MOD (1/3ohm 100watt resistor mounted to aluminum engine mount and spliced into the fan wiring coming from the fan. YOUR LOW SPEED FAN RESISTOR was DEFINITELY BURNED OUT it is a VERY COMMON AND CONSISTENT PROBLEM that most minis suffer from, the resitor bypass replacement is another very good Mod as the factory resistor assy and most aftermarket ones will burn up hence the BYPASS MOD solving the issue. there are youtubers with DIY VIDEOS. takes 15 minutes. try replacing the resistor you pictured with a 1/3ohm 100w. I couldnt see the resistors value in your picture Your new fan assy resistors shouldnt fail for several months i would think.
And you can wire a Fan relay bypass toggle to activate either stage at your will. Involves wiring a toggle to ground and running a wire to the fan relays. Using a test light to determine the ground points and connecting the wire to the terminal as you push the relay back into the socket.

I have an aftermarket temp guage installed and now run consistenly between 85 and 99 degree celcius. Stage one ECU PROGRAMING ACTIVATES LOW SPEED STAGE ONE AT 105C AND HIGH SPEED STAGE TWO AT 112. running up into these temps repeatadly over time cooks gaskets and seals and rob the engine of longevity and causes the overheating issues. Just be sure yoyu have no blockage and have bled the system of any air pockets AND ALWAYS ALWAYS USE BLUE COOLANT .....no other color than BLUE.... PEAK BLUE for european vehcles is way cheaper than MINI branded OEM and is just as good. But BLUE is the most important thing Each Color coolant is a different formulation for engines with different desgn parameters and materials used to make them ...

I also went with an aluminum Billet expansion reservoir since plastic sucks and is just used to save money in manufacturing costs. A pressurized closed coolant generates alot of pressure, the plastic res. literaly get blown like a baloon and eventually fails at the seams..... replacing the pwr steering res to Alum as well at the same time is also a worthy mod ..... Both easily sourced on amazon along with the Gold Aluminum Fan resistor (1/3ohm 100w resistor encased in gold anodized aluminum)

and to monitor the whole thing an after market digital coolant temp guage in one degree celcius increments to keep it running between 85 and 99 degrees then switching the toggle on when it pushes the 100 degree C mark...... My own preference for operating temp. the desgned operating temps are engine killer and rob longevity keeping temp under 100c solved all my issues and dramatically improved engine performance..... 180 degree them!!!!!! Toggle switch MOD. and a REAL guage ..... ignore the Chrono guage, its ridiculous and is programmed by the ECU to display the temp on a curve ....What car can withstand or even has ever approached 320 that guage is not showing you whats really hppening ..Install your own temp sensor just behind the upper hose connection and get a digital single degree increment guage and see the real temps in real time. The 180 degree F thermostat will help alot. and the toggle with let you determine the operating range . I have my toggle on the high speed fan cause it works very quickly to cool things down and i am running AC frequently hyere in CA which also solves any heat issues ... I solved all my similiar issues with my noted Mods but every car is different and overhauling and checking all possibilities is always the best bet.
I also reworked all the hoses under and to the side of the Airbox (I replaced a plastic T fitting that feeds the Expansion and trans oil Cooler and heater., i have an 6 speed Agitronic(not CVT) which necesseitate a trans oil cooler in the Coolant flow besides the heater core ......IT is a HOSE NIGHTMARE design but I improved flow by cutting out some of the stuff and rebuilt with aluminum fittings and silicon hoses...... it was a real pain and had to fine tune the hose clamps until i had no leaks but once i got it right the engine rewarded my efforts with really great performance and reliability.....

It is possible and you will figure it out and when you do it will say thank you..... just my two cents after alot of research and wrenching
FYI.....
06 MCS, K&N CAI (DIY), Yellowspeed Oversized Intercooler, One Ball Exhaust mod with no welding, ENA Performance Coil with 10mm cables and iridium plugs, Manzo Lowering springs, godspeed front end links, Doorman rear lower ctl arms (camber arms), Drag wheels w/Forceum HENA 205/45/r16, CravenSpeed Tower defenders and DIY tower Brace, and some other mods for ***** and grins... 15% pulley and retune upcoming......

I have a bentley manual (PDF) a haynes manual and a Haynes OBD!! ....... no replacing being an informed wrencher with as much literature at your disposal before doing any work.... IMHO

 

Seems like air in the system, especially if the air from your vents goes cold…

the s model has two bleed screws iirc. Does the non s? I’m sure the bleed procedure is in TIS and Bentley.