Why should we consider the output of the AES s-box as a “sensitive value” in relation to side channel attacks? In general,  we consider the output of s-box in AES as a “sensitive value” in order to perform a side channel attack. I would be very grateful if you could explain to me the reason of that consideration. Why should we consider the output of the AES s-box as a “sensitive value” in relation to side channel attacks?
I would like to complete poncho's answer by mentioning a point that has not been discussed here.

On top of being an internal state which directly depends on plaintexts (resp. ciphertexts) and the key if you attack the first round (resp. the last one), making hypothesis on S-box output (resp. input) allows to exploit the non-linearity of this operation. Indeed, due to this property, passing the hypotheses through the S-box operation ensures a good distinguishability between the correct and incorrect key guesses.

For example, let's say you are performing a correlation power analysis (CPA) by considering the Hamming weight as leakage model. Then if you choose to focus on the round key addition (which also constitutes an internal state which directly depends on plaintexts (resp. ciphertexts) and the key), then all your key hypotheses which have a small Hamming distance with the good key will also result in a high correlation coefficient. The S-box non-linearity offers the  advantage to reduce the peak for all the wrong assumptions.