Flex temperature isn't actually "fooling" the FADEC, right?
There are at least two misconceptions in your question:



The way I understand flex temperature is: It's a user interface for pilots to request a derating of engine power to reduce wear on the engine.

This is not quite correct. A derating would imply that the engine thrust needs to remain at the derated level after an engine failure. A FLEX or assumed temperature method takeoff allows setting higher thrust in case of an engine failure, so it's not a derating. See also What is the difference between Assumed Temperature and Derate takeoff thrust?



If the FADEC actually thought the air was less dense, then if anything it would increase engine power, not derate it.

This is not correct when the engine is temperature limited. The hotter the air, the hotter the EGT at the same fuel flow. Therefore, the FADEC has to reduce fuel flow to the engine for higher air temperatures, which reduces thrust.


Consider the following slide from this Airbus presentation:

You can see that the engine is flat rated (see What is a flat rated engine?) below a temperature Tref, which means the engine is not limited by EGT and can provide the same thrust regardless of temperature (see e.g. this answer for other limitations). If the temperature is higher than Tref, the FADEC has to reduce fuel flow to keep the EGT below the limit. This results in less maximum available thrust.
By entering a FLEX temperature, the FADEC will reduce fuel flow to the engine as if the actual temperature would be that high. We can argue about the semantics of whether or not this is actually fooling the FADEC, but the fact is that the FADEC will reduce the fuel flow accordingly. This will then actually give you a lower EGT than the maximum because the actual air temperature is colder and therefore reduces engine wear and tear.
The FCOM contains some additional information about what the FADEC does:

The FADEC system performs the following functions:
Control of gas generator

control of fuel flow
acceleration and deceleration schedules
variable bleed valve and variable stator vane schedules
control of turbine clearance
idle setting

[...]
Power management

automatic control of engine thrust rating
computation of thrust parameter limits
[...]


(Airbus A320 FCOM - Power Plant - FADEC)

When the thrust lever is in the FLX/MCT detent:

On the ground
The engine runs at the flex takeoff thrust rating if the crew has selected a flex takeoff temperature on the MCDU that is higher than the current Total Air Temperature (TAT). Otherwise the engine produces Maximum Continuous Thrust (MCT).



(Airbus A320 FCOM - Power Plant - Thrust Control System)