In your initial post, you said a float 'VALVE' turns on the pump.
If a float valve is utilized, the valve will open when the water level is low in the aerator so as to cause pressurized water from the pressure tank to enter the aerator. This will in turn cause the system pressure from the pressure tank to become gradually lower, thereby causing the pressure switch to activate the pump when the system pressure has fallen to the PS activation setting.
If you actually mean a Float 'SWITCH' is utilized, then low water level in the aerator will cause the float to activate a switch which will directly activate the pump. When a float switch is utilized, a pressure tank and pressure switch will not be required.
The float still controls the pump, but I’m not sure how the pressure turns on the pump. No additional electrical was installed. And what pressure is the pressure tank monitoring?
A pressure tank supplies system pressure while the pump is not running. The pressure switch monitors the system pressure so as to activate the pump when the system pressure has been reduced to a specific pressure, and will shut off the pump when the system pressure has been increased to typically 20 psi higher that the activation pressure.