I think a point you are missing is that FADECs do not adjust the thrust to achieve a target level of aircraft performance - the engine doesn't really care about that.
Engines are designed to give the commanded thrust, not to modulate the thrust according to some "what the pilot needs" process.
It might be easiest to first consider that you can flex with a NON-FADEC engine too, and work through that more simple system example. then the FADEC case is really just the same.
So on a non-FADEC engine and aircraft, it's a cold day and you have bags of performance "in hand" - the runway is much longer than you need, there are no obstacles etc. So you don't need all thr thrust you have today. But how much can you safely reduce it by?
Well, you look at your performance charts and work out "if it was this (hotter) temperature then i would just be able to takeoff safely". That then tells you how much less thrust you can safely use.
Now you look up your manual thrust setting table, and find on that hot day you'd set say 92% N1 instead of the 95% you can use today. So you "flex" to the temperature that just makes the takeoff safe, and set the corresponding thrust, and away you go.
The only difference in the FADEC case is that you don't have a manual to set the n1 - the FADEC calculates it. So you need to tell the FADEC "hey, let's pretend its hotter today" by giving it the flex temperatuire, and everything else works the same.