I think you may be getting confused because the latest bikes need some pressurisation procedure to get coolant in the system. Cooling systems aren't like your tires!
Our cooling systems have a reservoir sitting slightly higher than the radiator cap, but the level in the reservoir is not necessarily the important true level in the radiator because you can get air or a large air bubble trapped. In my experience this is more likely to be at the top of the right radiator where there's no cap.
The trick is to remove the left rad cap (carefully!) when the system is warm (but not hot!) and burp or squeeze the bottom hoses. I've also disconnected the top hose where it runs to the left rad and 'injected' coolant into the right rad with a baster, whilst burping the hoses. Before removing the radiator cap, pinch the small reservoir hose coming into the left rad with a pair of grips to close it off. This stops the reservoir coolant pouring out and leaving air in the pipe to be purged out.
Finally, there will be some air left at the top of the radiators. Top up the reservoir to normal and run the bike up to 'fans on' at idle and allow to cool 1/2 hour whilst watching the reservoir. Make sure the front of both radiators feels hot at 'fans on'. If the right rad feels cold, then you have a faulty thermostat, blockage or a big air bubble in there. The reservoir level will rise, sometimes air bubbles will work through, then as the motor cools the level will drop. Do this a couple of times. You should do this every time you remove the cap (and let air in). There have been reports of pressure caps going faulty and not sealing properly so the system won't pressurise when hot and pukes coolant out of the reservoir - a cheap fix to replace and good insurance.
The problem with the K bike cooling system, is they didn't put a bleed hole in the thermostat under the right rad. This means when the motor is cold and it's closed , you cannot get coolant across to the right rad that you pour in through the left rad pressure cap or reservoir.
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