For singing to begin, the vocal folds vibrate against each other. The sounds are amplified and changed as they pass through the vocal tract, out of the mouth (and/or sometimes the nose), and into the world. So far, true for all singers.
They vibrate because air is blown against them. Also true for all singers.
These bare facts about phonation and breathing are less refutable than everything else. If we call the source of sung sound, the conversion of air into vibrations at the folds, the “center”, then the further you get from the center, both up and down from the larynx, the more you will see individual differences – in the structure of the singer, the sounds they make, and in the way they use their bodies.
Insisting on one particular way of breathing for all singers, with all the same outward signs, won’t work. The same breathing actions will look different in different bodies, and different bodies may require different actions.
The shape and size of the vocal tract is also different in every singer. There are small tracts with large larynxes, and large tracts with small larynxes. Wide, long, skinny, short, and fat necks. Mouths of all sizes and shapes. We’re each playing a very different horn!
The items above involve the parts of the singer’s body that are progressively farther from the source of the vibrations. Beyond the physical, we move into the intangible differences: the singer’s concepts of the sounds they want to make, their manner of pronunciation, their kinesthetic sense, what they are feeling emotionally, their interpretation of the song, and all of the other psychological stuff.
You can teach similar principles to everyone, but you can’t 100% teach the same ways. I recently read a book about singing that guaranteed that if you do exactly as the author says, you will have success. The author says that if there is a problem, it is due to failure on the part of the reader. Oh my! I have heard another teacher claim that “in every single case, ___ing the _____ has helped my singers.” (gerund phrase omitted to protect anonymity), when my experience totally disagrees with that statement.
If you are a teacher who insists on a certain rules for singing, even in the categories of technique and practice that are far from the center, you may crush a student’s progress. Teachers with waiting lists or who are on the faculties of competitive schools tend to be less aware of this, because those who fail will quit, and someone more successful will eventually take their place. A teacher in these situations is more likely to keep their particular ways intact and feel that they are “proven”, regardless of casualties.
Meanwhile, we independents over here in the private marketplace take in the wounded and confused, and get to work on the issues brought to us, working together with the singer to help them build their own process, whatever it may be. When we fail to help, the persistent student should move on until they find someone who can. If you want to help singers more than you want to propagate yourself, you will do okay.