In classical physics, the Lagrangian formulation is merely useful for some calculations, and can be fairly viewed as a mathematical trick that is equivalent to using Newton's law directly; similarly in E+M it's a compact and often useful way to express Maxwell's equations, but not particularly fundamental.
When you reach quantum field theory, though, it's all but impossible to do anything without a Lagrangian/action, and theories are almost always described in that form. String theories are the same. General relativity is not always presented that way, but gravitational dynamics are stunningly simple in that form: in a vacuum, nature abhors curvature.
Principles of least X are fascinating and incredibly useful, and the ex-physicist in me believes that they describe the actual universe so concisely that they are truly fundamental in some sense. But you have to really dig in to get any sort of appreciation for that, and I agree that this article didn't go anywhere near deep enough.