Emitter is unlikely to die on its own. The driver might be toast. In that case, a new emitter wouldn’t fix it. It’s not straightforward to test a 6v emitter and don’t ever try to connect a 6v battery directly to it otherwise it can easily burn out in a flash. If you do have 6v power source, use a series resistor to limit the current. Something like a 10 ohm is a safe bet.
Also if the driver is dead, there is not much you can do. The host and driver are proprietary so other drivers won’t fit. You can salvage the battery, emitter / MCBCP and the optic at most.
The white wire seems to connect to the temperature sensor on the MCPCB so could be a temperature feedback for the driver.
I guess it’s just one of those infant mortality things for electronics. When you get it, stress test it to make sure it works.