The first thing you learn about the Dirac equation is that it provides a relativistically-correct quantum-mechanical description of spin-1/2 charged particles, e.g. the electron.

Then, it seems that it's at least implied that the Dirac equation completely describes the interaction between multiple Dirac particles (e.g. electron-electron)...

However, I don't think I've ever seen it spelled out explicitly like this (from Houghty - Lagrangian Interaction):

So am I reading that correctly? Do the Maxwell equations literally follow from the Dirac equation? I've seen them put together in coupled equations of motion (e.g. the Dirac current = source for Maxwell field, Maxwell field = external potential for Dirac field), but I always took those as empirical rather than necessary deductive fact...

It indeed appears that way. The so-called classical fields which take values in a complex vector space (complex scalar field), or generally in an involuted Grassmann algebra over the reals (the Dirac fields) have the U(1) / phase symmetry built in. Gauging this symmetry with the purpose of describing possible interactions leads to the existence of a theory of U(1)-valued 1-forms which are interpreted as the potentials for a free e-m field. Not gauging the global symmetry, well no e-m interaction between the scalar or 1/2 particles. That's easy.

Surely, one can consider the 'matter' fields without the involution property, so no global symmetry to gauge, except perhaps for the Poincare/Lorentz one which would then describe the gravitational interaction between these fields at least a classical level (level which doesn't really exist for any non-gauge fields).

As a conclusion, the need to develop interactions between matter fields (charged or not) leads to the possibility of gauge theories (the simplest being the e-m theory) which would 'mediate' these interactions.
But the Dirac field & equation can be analyzed on their own, as theories of free particles.

You just blew my mind. Any suggestions for reading more on this topic? I've got a few books on basic gauge theory, but they seem to be 100% focused on the gauge fields themselves, rather than how they could arise from matter fields (or even how they would interact with matter fields given that they have arisen somehow). In particular, anything that focuses on the classical field level (rather than delving into second quantization -- which seems inevitably to distract at least the higher-level theoretical dynamical understanding) would be great.

I've seen that term bandied about in a lot of different contexts -- the most only consistent actually has been as a name for the electrical "correction" to the canonical momentum (e.g. p -> p + eA).

It still saddens me that it seems the only place to read stuff like this is in an aside in a QFT text.

I can't think of something definite. I've read chapters from certain books and tried to capture the ideas which would give me a decent overview, which I probably have.