My 2 cents- checking one’s pitch against a tuner IMHO is to get the mouthpiece adjusted where the embrochure/lip control of the pitch has the most relaxed “natural” reference position to achieve playing the range of the notes of the sax “relatively” in tune. But an accomplished player has to have trained muscle control of air delivery, throat muscles, oral cavity/jaw/tongue muscles, embrochure muscles, to be able to “auto-tune” on the fly. Most of the saxes I have played are plagued with lower register notes that are on the flat side, and higher register notes that are on the sharp side, and various other notes throughout the spectrum that un-characteristically sharp or flat to the adjacent notes above or below them.
When someone checks his pitch against a tuner, they are very likely subconsciously making the corrections to center the needle- be it making their air delivery very precisely controlled, their embrochure in it’s most natural position, etc..
Now put that person into the live dynamics of playing music where all the other moving parts have to synchronize, expression with all the embellishments to be added, the air delivery has to be adjusted to control soft and loud notes. The way you end notes can make them go sharp as we have heard Johnny comment on time and again.
Then the muscles can begin to fade due to a long passage with no break, and your ability to be able to keep that pitch dialed in fails.

So checking one’s higher notes to a tuner and getting the impression that hey I played them in tune, doesn’t mean that one will be able to play them in tune throughout a challenging piece of music, especially if your chops or air supply can go the distance…

When I performed at church last week I had tuned to the piano at home the day before. I was only able to tune the notes of the left hand G# G A A# B C# C ect. for I needed the right hand for the piano. I adjusted the mouthpiece a few different times until I felt comfortable with all the different notes and I left the mouthpiece on the sax. went and played the songs at church the next day and was amazed that the notes I played sounded so in tune, they really sang out. was it a fluke I don’t think so. but most of the notes were in the mid range the highest was a second octave A. That A and and The E are what caught my attention and as I was playing and I said to my self perfect. I had played the songs at home with out using the backing tracks. I also stood close to the piano and could here his monitor witch I am sure helped.