Check (depends on your F/W rev) Setup Menu 04. or Settings Menu 05. "Cold End" is not scrambled or reading a silly temperature. Wand plugged in, it should read room temperature if the CJC thermistor bead in the wand is wired properly, regardless if a tip is plugged in.
Unplugged wand: The controller should display ERROR and the LED will only be lit if a shorted MOSFET.
Since the thermocouple input is floating and picking up noise, the controllers freak out and display ERROR or flicker between run/error mode, this is normal.
Otherwise I would suspect the thermocouple op-amp is not working.
You can unplug the wand and connect a 10k resistor to GND from heater connector pin at the top T12+ or pin 5 on the GX12. With that very low mV input, a low temperature should be displayed and the controller will think it's ok to run and turn on the heater (which is all unplugged).
If you still get that silly high temperature reading, check the SGM8551XN op-amp input (pin 3/LED) is around 0V and the output pin 1 is low under say 250mV.
edit: op-amp output is about 5mV/°C so 350°C is about 1.74V to the A/D. Room temp 25°C should be about 125mV
"OLED-V3.0" schematic is basically the
same circuit as V2.1S, with a different 6-pin OLED, no pump control mosfets, different reference designators etc.