That sensor box thing on the top of the fan shroud is for the automatic recirculation. Ac should still work without it. My ac system has a fault code for that sensor, auc sensor.

Most ac problems are caused by loss of refrigerant, the system has pressure sensors that will stop the ac from trying to run when there is not enough pressure, it does that to prevent damage to the compressor.
Check for leaks. They will look like just slightly oily spots on the compressor, lines, evaporator, drier, sensors, something will have just a slight oil residue. They can be very difficult to track down.
That is why ac leak tracer dye is so useful.

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