map analysis

In a circular accelerator, if you look at a single beam position monitor (BPM), or say, a couple of them, to give you phase space, you essentially looking at the properties of the one and many turn behavior of electrons. The value of the Hamiltonian -free description, the map description, is that it treats the machine as a whole, without necessarily thinking of how that map comes about.
This is analogous to solid state physics and scattering analysis. You define various quantities of the material that can be probed via scattering stuff off it. You don't need to know the position of every atom in the material in order to define these quantities.

Why has this approach not developed further in accelerator physics? Because there aren't enough accelerators. Each one has its own peculiarities, and so people think more about how to change those peculiarities and effect certain global behaviors rather than be more creative in defining the global behaviors. It would be like analyzing ten space ships, each of which was rather different from the other. One would probably not extract a general theory of space ships out of this.