I've been browsing a bit and enjoying reading some papers. I found this paper on interpretations of the Dirac equation here (M. Valenti, 2008) which I've skimmed a bit. He talks about using QED to describe the Hydrogen atom, which I'd be interested to understand better. It does seem that mostly QFT calculates S-matrix type stuff, and bound states are more foreign. If NR QM and the Dirac equation really come out of QED, then it should be able to deal with bound states. I vaguely remember something about "resonances" (related to the complex poles) of the S-matrix being the bound states...
Maybe too hard to understand right now, but interesting stuff anyway.
(added... Ok, here's an interesting quote related to this reductionism, model building stuff:
In this way we are not restricted by Haag’s theorem – and so we can retain the concept of quanta in the description of interactions – because, from a physical point of view, the Lagrangian of quantum electrodynamics does not provide us (contrary to what from a mathematical abstract point of view might appear) with the possibility of describing a system of (undifferentiated) interacting Dirac and Maxwell fields, but with a way of developing models that describe in a limited way the interaction between the fields.Valenti, p. 14)