Engine coolant radiator fans

Six cylinder front wheel drive automatic 130,000 miles.

I am the owner of this vehicle and I am relaying the problem as my mechanic has described it to me:

The hi-speed radiator fan is not coming on when the temperature goes over 96 degrees centigrade. (The low-speed fan is operating normally.)

The engine coolant overflows as a result.

We have installed a new engine temperature sensor, a new thermostat and even reprogrammed the engine control module.

What we do not understand is that when we unthread the engine temperature sensor from the housing, leaving the electric connector hooked into it and with the ignition key on, when we place the sensor in a bowl with water and heat it to 96 degrees centigrade the system responds and the fans come on like they should when the sensor is threaded into the thermostat housing.

1) Engine overheats.
2) Coolant overflows.
3) The high-speed fan has been determined to not activate.
4) When high-speed fan is manually activated, engine does not overheat and coolant does not overflow.
5) a) The ECM (Electronic Control Module) controls fan activation, is this not correct?
B) ECM has been re-programmed.
6) New thermostat has been installed.
7) New engine temperature sensor has been installed.
8) Intake manifold is functioning normally.
9) This we feel is the main issue. It seems to be clearly the main issue. Please advise if we are misguided on this:
"The engine temperature sensor screws into the housing. On the other end of the sensor are two wires that connect to the ECM. When sensor is screwed into housing and its wires are connected to ECM, high speed fan does not come on and coolant overflows. When sensor is unscrewed from housing and heated to 96 degrees centigrade in a pail of water, high speed fan comes on and coolant doesn't overflow! The two wires that connect the sensor to the ECM have been checked and are transmitting proper voltage."

A thermo switch has been installed temporarily so that the vehicle can be driven.