[adrianratnapala]

It's the focus on path integrals that obscures the fact that the Lagrangian is an operator. Except maybe for convenience, there is nothing about such equations that is unique to the path-integral formulation.

The fields, the Langragian, and the scattering matrix are all operators; but of course their matrix elements are numbers. The path-integral formulation is a good -- and I think physically well motivated -- trick for calculating those matrix elements in terms of merely fields and their associated Largrangians.