The lift formula is in fact a definition. It's the definition of the coefficient of lift. There is no knowledge implied in a definition. You may define whatever your want.
The knowledge comes when you make experiments in a wind tunnel. Then you can see that the coefficient of lift, at usual GA aviation speed & pressure, does not depend on speed, nor on temperature, nor on pressure. The coefficient of lift only depends on the angle of attack and on the shape & surface of the airfoil. That's an experimental knowledge, and it's enough to built safe aircrafts.
Most of the time in history, knowledge comes first with experience, then a mathematical model is built and some theoretical knowledge can be built on the basis of the previous experience.
Most explanations of lift in pilot's manual don't explain anything (see
https://www.grc.nasa.gov/www/k-12/airplane/wrong1.html,
https://www.grc.nasa.gov/www/k-12/airplane/wrong2.html, etc.).
The best explanation, to my point of view, is to explain the actual experimental protocol that led to the properties of the coefficient of lift.