As an aeronautical engineer, we do know why airplanes fly. There are multiple ways of analyzing them, but it isn't F***ing magic. I really hate stupid articles like the one referenced. Apparently the people who wrote the article don't know a damned thing about circulation, or the Kutta-Jukowski theorem. Different "approaches" offer different data based on what is being sought; as they noted, Bernoulli and Newtonian Mechanics. They're both valid. They could also perform a Trefftz plane analysis,etc.
BTW, for simplicity, think of it like this. F=MA. The airplane stays up by throwing air down. Equal and opposite reactions. The different mathematics/physical approaches are just to reveal different data sets based on what we're seeking to understand about the vehicle's performance.