Why does one range burner only turn on when another is turned on?
You clearly have a wiring problem.  It sounds like you have lost one phase to the burner that has the problem.  When you turn on the second/support burner, that switch is back-feeding power to the problem burner switch. 

Also inherent in these kind of problems, you may be getting the 240-volt to all the switches but somehow the wiring connections or wire leads have lost the ability to carry a load. Look for burned and or worn leads and connections.