As a former saxophonist I know for a fact that the fingering of the clarinet and the transverse flute are very difficult compared to that of the sax… That of the sax (a few sharps aside) is very similar to that of the recorder… the big problem of sax is to bring out a sound rich in harmonics, the reed which must be of the right hardness compared to the opening of the mouthpiece… in short, a lot of “problems”… lips dry from the cold, from too much heat. …and the trumpet is even worse…
… and then, unless you play in C, the sax is in Bb (Soprano and Tenore) and therefore you have to have a score shifted in key and this is also a further problem especially if you play with the “damned guitarists” 
who play everything in E which for the sax becomes F# and therefore a scale that has a more complicated fingering…