From learning about the Lagrangian lately it seems that that it can underlie so many phenomena that it must be the unifying concept that underpins all physics. I often hear that physicists are searching for the theory of everything, why is the Lagrangian approach not considered to be such a theory?

On a related note, can the Lagrangian (and/or) Hamiltonian approach to describing a system be considered the ultimate test of validity for any physical theory? In other words, is it necessary that every fundamental physical theory can be derived using the Lagrangian approach?