Historically, the reconciliation of quantum mechanics and relativity led to the single-particle Dirac’s equation

$$(i\gamma^\mu\partial_\mu – m)\psi = 0.$$

This last allowed to explain correctly the fine structure of the spectrum of hydrogen atom and to predict the existence of anti-particles. However, it also lead to inconsistencies: negative energy states and a non-positive-definite probability density. These inconsistencies motivated in turn the development of a full relativistic quantum theory of fields: QFT. However, bound states - such as the hydrogen atom - are not easily treated within QFTs. The Bethe-Salpeter equation is the most orthodox tool for discussing the relativistic two-body problem in QFT. In particular, the hydrogen atom is usually treated in an approximation where the proton is treated as an external Coulomb field.

Given all this, it seems reasonable to expect something like the original Dirac Equation with a Coulomb potential to be recoverable from the standard model through the Bethe-Salpeter equation or analogue methods. As far as I understand, this is in the same spirit of what happens in chapters 13.6 and 14 of Weinber Vol 1. However, I am unsure my reading is correct.

In summary, is my exposition correct? Is the expectation of such a derivation justified? Has it already been done and if so where? And with what subtleties?