Good and bad reasons to explore the parameter space of effective Hamiltonians

Too often I encounter papers or people that describe some detailed study of a particular region of the parameter space of some effective model Hamiltonian (Hubbard, Holstein, Hofstadter, ...) and I am left wondering, "why are you doing this?"

The most cynical answers I sometimes fear are "well, it is there" [just like Mount Everest], "no one has done this before", and "I can get a paper out of this."

It is not hard to find regions of unexplored parameter space for most models. One can simply add next-nearest neighbour hopping terms, change the lattice (anisotropic, triangular, Kagome, honeycomb, ...) , or add more Coulomb or exchange interactions, or add a magnetic flux, ....
The list is almost endless.
But, so what?

Here are a few good reasons of why exploring a particular parameter regime may be worth doing. Only one of them is sufficient. Often only one may be true.

In this parameter regime:

A1. There is an actual material that is believed to be described by this Hamiltonian.

A2. Given the inevitable uncertainty in the actual parameters for a specific material it is worth knowing how much calculated properties change as the parameters are varied.

B. There may be a new phase of matter or at least qualitatively different behaviour.

C. One can gain physical insight into the model.

D. A particular analytical approximation or numerical method is known to be reliable.

E. Provides a good testing ground for the reliability of a new analytical approximation or numerical method.

I welcome comments.