That the exponential is the function which Putzer's method computes is largely irrelevant. He probably had a need for that particular function, for which the differential equation allowed a self-consistent presentation of the method. However, the survey article written by Cleve Moler and Charles van Loan [19], ``Nineteen dubious ways to compute the exponential of a matrix,'' concentrates exclusively on the exponential function while examining the strengths and weaknesses of a goodly number (19, to be exact) of schemes. Of course, one of them was the one which Putzer had published twelve years earlier. Given that differential equation integrators, using Runge-Kutta methods and others, show good performance, and that Putzer's process implies a sparse matrix, it counts as a viable way to get a matrix exponential.
The title of the article reflects the computers and program libraries available at the time it was written; the conclusion reached was that all of some seven categories of programs had strengths as well as weaknesses, leaving the choice of a particular method to depend on circumstances - none was clearly and unfailingly superior to all the others.