Wax increases in size as it melts (as opposed to water) so it would make sense that a small cylinder filled with wax would cause a small piston to be pushed out as it heats up, and that piston would be connected to the needle. When you turn the motor off long enough for the wax to solidify, it would then contract and pull the needle back to the starting position. Presumably there's a coil wrapped around the wax cylinder that causes the heat to melt the wax.
The way I have tested it before on my old Vitality is to connect a small tube to the channel that lets the additional fuel pass through (with the autobystarter in place), and blow. If you can blow air through, the autobystarter is open. Then, connect the autobystarter to 12 v for a few minutes, and if you can still blow air through the hose, it is not working properly.
It doesn't sound to me like the autobystarter is the problem though, if you could actually see the needle move, but doing the blow test may be a way to see if there's a blockage in the channel.