TAT is a very important indication.
Aside from letting you know when you must have the AI on, it is one of the elements (along with pressure altitude) used in calculating the engine power setting eg Maximum continuous thrust.
Of course in the modern jet, this is often done automatically.
As for when to turn off the AI. Of course that would be in accordance with the AFM. One does not have to search for ther SAT if one has the TAT and speed (which gives you the temperature rise) but it is easier.