Ok I think, thanks to Zips pictures I have worked out the circuit. Am away from home so please excuse the picture.
Am not certain about the transistor, the marking looks like 6C and that seems to be a BC817, using it's pinout the circuit is unusual but makes sense.
So we have a series protection diode from the +ve, this feeds the collector of the transistor. I am assuming the other diode is a zener, unknown voltage but would guess 4 to 6 volts. The zener is fed through a 1k2 and will hold the base at a constant voltage. The voltage on the emitter of the transistor will be about 0.7V less than the base and is used to drive the two LED's each through a 68 ohm resistor.
So SCX seem to have put in a simple voltage regulator for the LED's. This should mean that the brightness remains constant over a range of voltages, so in analogue mode less variance than a Scalextric car?
So bear with me while I think out loud.
The transistor is rated at 250mW and 500mA. The current rating is fine. Am not sure how many mA each LED will take, but let's guess 50mA so 100mA in total.
So if the emitter voltage is say 5V and we have a 15V supply then the transistor will have 10V across it and is passing 100mA. So power dissipation is 0.1A * 10V which is 1W. Too much.
So perhaps I have assumed the wrong current for the diodes. So looking at it from the other end. The max current for 250mW is at 12.5mA each, 25mA total. Power dissipation now .025A * 10V = 250mW which would put the transistor at it's limit.
So if this is a BC817 it is not suprising that it fails, if there is a suprise it's that it survives even with 14.8 Volts. It could be I am still making a wrong assumption about the LED current. To be safe it would need to be a max of 6mA each.
Also possible that the device is not a BC817. Could someone make a couple of measurements on a good board. With the car at max throttle measure the voltage on the input +ve, then on the three connections of the transistor which is the black square in the middle of the board. Put the negative meter lead on the Negative (RH 0V) input strip.
Rich