.... it's not an instrument you can pick up and learn on your own.
That's not entirely true. I have had a tenor sax for just over a year and have had great fun flying solo from the start. Admittedly, I still have some unexpected squeaks, particularly at the octave change, but getting to grips with the basic navigation isn't too difficult if you have played a whistle, recorder or some such instrument. I still have a long way to go but, as it is strictly for home consumption, I'm in no rush.
I'm sure that good sax players feel the music as much as play it – it's that sort of instrument. Hopefully, a good sax teacher understands this and would not stifle your individualism by insisting on a "correct" method.
The only instruments that I have had lessons on are the piano and violin, and they are the instruments I am least relaxed with.