I really don't see how the thermostat has anything to do with the fan not working. They are two independent phenomena. I do understand that hot coolant must be flowing past/around the thermoswitch for it to turn on, but the op seems to have confirmed the left radiator gets hot.
If the water pump weren't working he would have other symptoms of overheating in extremis as well as late or no actuation of the fan. He didn't mention that.
When burping with the cap off, you can see the coolant flowing according to revs, thus confirming the pump is working and thermostat is open.
It keeps coming back to your fan not turning on when the temp reading on your gauge cluster says it should.
I would primarily suspect an issue with the fan and/or its associated wiring.
At a long shot it could be as someone mentioned, that the reading displayed on your instrument panel (gauge cluster) is erroneous misleading you to think the fan should be on when it shouldn't, though I seriously doubt this. Still, if you wanted to check out that possibility, it is possible to check all the associated wiring without removing the airbox and throttle body, but yes, to check the sensor itself you would have to dig in to the sensor itself on the back of the front cylinder block (in the V of the motor). I sincerely think this would be barking up the wrong tree.
You may have an intermittent fault in either the fan wiring or the fan itself. These intermittent faults are a royal PITA. Almost impossible to reproduce so as to pinpoint the exact location of the fault. Often they only raise their head in certain circumstances, such as when hot, or due to vibrations or wires moving in the wind when in motion. This is particularly pertinent when checking the fan operates by ground directly and bypassing the thermoswitch with a paperclip or wire, as we tend to do this when the bike is cold, so as not to burn ourselves, but thus not factoring in heat-provoked faults.
Unfortunately, often the most effective way of remedying such faults is to replace parts. This should always be done one by one if you really want to know which part is to blame. If you have just had enough and want it fixed you can replace all suspect parts and swallow the cost.
I understand you have already replaced the thermoswitch, so unless by freak chance you got a faulty one off the shelf, that item should now be ruled out as the culprit.
Perhaps replace the wiring in case there is a small crack or other imperfection unseen to the naked eye. Then replace the fan which might just be on its way out, occasionally and intermittently behaving or misbehaving. I would suspect the wiring first.
I also don't see what the radiator cap has to do with fan operation. You report both radiators are heating up. I assume too hot to touch after a while. I guess if the cap is faulty it could be letting air in but you would also have the issue of the coolant boiling and the overflow tube spewing coolant onto the ground, which you have not mentioned.
Otherwise you just have a stubborn bubble at the thermoswitch location and will have to get creative with dislodging it.