In one of my first ever sax lessons, my teacher got me to stand with my heals and shoulders and back of my head touching the wall.
In that position you need your neck strap high enough that the mouthpiece can be just tipped into your mouth by your right thumb (ie C# fingering)
My teacher pointed out that almost no-one has their sax high enough! You may need to loosen the neck strap very slightly from that position, but that is how high the sax actually needs to be.
All the weight is on the strap, the right thumb just stabilises the sax a little.
That said, if I am playing a long (in time) C#2, then I actually play a low C#1 and add in the octave key.
The C#2 with no fingers is not a pretty sound at the best of times!