If you play a middle C, then you can place the mouthpiece anywhere on the sax neck and play that middle C in tune, by just adjusting your embouchure.
However there is really only one place for the mouthpiece depending on the neck dimensions and the type of mouthpiece. Once you find the right place, mark it on the cork, then every day depending on room conditions all you need to do is shove it in a fraction or pull it out a fraction from that pencil mark.
A Pro looked at my mouthpiece and told me it was in the wrong place on the neck after i told him i had tuning issues. I had to move my mouthpiece in by half an inch, this took me several weeks to change my embouchure to play in tune. Each day i shoved it in by a tiny bit more until i could finally play in tune with the mouthpiece a half an inch inwards.
The one thing i noticed later on, was when you did Johnny’s demo of comparing the lower overtone F with the middle F, they sounded not just more in tune, but their tones sounded more identical. Once this occurs all the keys in all three sax registers will be easier to play in tune.