The Dirac equation works well for predicting the spectrum of the hydrogen atom, famously incorporating relativistic effects like fine structure. Yet, there seems to be a sense in which this is accidental. Specifically, let me quote the following passage from Sidney Coleman's recently published lecture notes, where he argues against the usage of the Dirac equation as a single-particle wave-equation (as opposed to a quantum field theoretic description):
"As a general conclusion, the corrections of relativistic kinematics and corrections from multi-particle intermediate states are comparable; relativity forces you to consider many-body problems. There are however very special cases, due to the specific dynamics involved, where the kinematic effects of relativity are considerably larger than the effects of pair states. One of these is the hydrogen atom. That's why Dirac's theory gives excellent results to order $(v/c)^2$ for the hydrogen atom, even without considering pair production and multi-particle intermediate states. This is a fluke." - Quantum Field Theory, Lectures of Sidney Coleman (Page 1).
Of course the hydrogen atom is very special, being integrable, and being one of the few exactly solvable systems in quantum mechanics.
But physically, what exactly is it about the hydrogen atom which makes the single-particle Dirac equation work, where we expect it fails for generic relativistic systems? What is the fluke here?
More generally, are there generic physical conditions under which we can expect relativistic results to be faithfully represented by simplified single-body techniques?