The bin-level sensor is really just a plain old thermostat that’s got a fixed cut-out range of 32-degrees ( the point in which it contacts the ice rising in the bin & shuts off the compressor)
Most Ice machines use a low pressure controller to regulate the compressor’s cycle along with ( typically) a high pressure cut off switch to protect the system from excess temperatures from say a blocked airflow ( or water) to the condensor ( or water cooled heat exchanger)
There’s usually a water inlet valve and/or a pressure regulator which may be temperature or pressure controlled to regulate the flow of water to the freezing chamber and/or the condensor heat exchangers ( for a water-cooled system)
So in summary, to make more ice faster, you’ll need...
A: more cold. ( a properly charge & controlled refrigeration system)
B: more water. ( adequate flow & inlet temperature)
C: more speed ( proper auger feed in the freezing chamber, belt slippage, etc.)
See? Easy peasy