Low-spin / high-spin arguments: why overlooking pairing energy?
Regarding your question about the non-linear change with ligands, I suspect this isn't that easy of a question to answer, but the data is probably out there somewhere.

Regarding the quote and question in the title, when these kinds of arguments are made, they aren't ignoring the pairing energy, but they also aren't really making any predictions.

Essentially, Ligand Field Theory (LFT) lays out a simple way that one can rationalize the geometry of a particular transition metal complex based on the energy of the d orbitals. I can see that you know this. What this means though is that when they observe whatever change happens for the iron complex, LFT says that this must correspond to some change in the energy of the electrons associated with shifting changes in the energy of occupying orbitals in a specific way.

That is, in LFT the only two terms that describe the energy of a particular electronic arrangement are the ones describing the potential energy of the electrons while ignoring their interactions, and the increase in energy from pairing electrons in orbitals.

So, when any geometrical or similar observation is made, there's really no other explanation than to say the a high-spin complex became low-spin because the orbitals split more ("considerbale change in LFSE") which guarantees this decrease in potential is larger than the cost of pairing electrons.

I agree that it's quite naive to always explain things in this way, but that's why crystal field theory and ligand field theory are not meant as real computational or predictive tools these days. They are useful for connecting geometry, spin, and energy as well as giving a first-order explanation of observations.

It's like when you'll sometimes hear inorganic chemistry students describe the behavior of a catalyst as being due to a "combination of electronic and steric effects." That's true... but those are also the only possibilities, so it doesn't tell you much.