Apparently your RC is 2002-2006 model on which the fans are activated by the ECU through a relay in the tail.
When you hit the Kill Switch Off, the ECU is not powered any longer so the fan's relay disengages. If you had the 2000-2001 SP1 model with the, bolted on the LH radiator, thermo switch, the fan should keep working with the key to ON and Kill Switch to OFF position.
According to the service manual, the temp's red light comes on at 124 degrees Celcious (255 Fahrenheit). But your 250 reading is quite correct I believe.
If the red light comes on, then you are seriously risking your cylinder head gasket's integrity at first and the whole engine's next.
I had the same problem a year ago which turned to be a very cheap coolant agent's fault. When I took a close look at the coolant's bottle, it said to withstand up to 102 degrees C (215 F). Then I flushed all out and replaced it with this
which is concentrated and if you dilute it 50-50 with destilled water it can deal with temperatures up to 150 degrees C (302 F).
You cannot imagine how fast the temp gauge goes down when the fans turn on.....I was amazed. No problem at all.
Some other, rather extreme, causes to rise up your temperature are
1) A faulty thermostat valve
2) Air in the coolant circuit
You should check those factors anyway.
The service manual will help you through the process