What about breathing? Doesn’t correct breathing play an important part in your ability to produce good tone?

Of course. But the importance of breathing in singing has been overemphasized by voice teachers for too long a time. Correct breathing is a by-product of good technique – just like one’s resonance quality is a by-product. You should never work directly at developing your breath unless you have a sloppy posture and a depressed rib cage (which collapses uncontrollably when you expel air). You indirectly develop the proper breath support for your tone as you condition your larynx not to move and your outer muscles to relax. When you use a speech-level approach to singing, everything, including how much air you use to move your cords, happens automatically.