A slightly different motivation for fourier integral operators and pseudo-differential operators is given in the first chapter of this book - Fourier Integral Operators, chapter V. W. Guillemin: 25 Years of Fourier Integral Operators. In this chapter, V. Guillemin presents this subject from the conormal bundles point of view and then shows how pseudo-differential operators are a special case of operators coming out of the Fourier distributions.
Section 3 of this chapter also provides a historic outline of this theory.
Interesting, Guillemin also shows here that pseudo-differential operators can be used to fine tune the theory of conormal distributions themselves -- giving a natural but unexpected application of pseudo-differential operators.
