Is this a type of formalism that extends across multiple field theories?

Yes, it underlines them, so to speak. You can even derive the standard quantum mechanics representation from it. It's also often clearer to operate on symmetries and prove conjectures for entire classes of objects.

SQMR: the classical algebra of functions of position, covariant for the group of shifts in space.

My understanding is that Lagrangian mechanics better accommodates for Newtonian/classical-scale dynamics because the difference in potential vs. kinetic energy is much more defined than at the quantum scale, where total energy is a more useful framing.

As a rule, you'd use Lagrangian over Hamiltonian whenever you want to keep the momentum of a system you're describing fixed to a value; convient from particle collision physics to path integral formulation. Hamiltonians are convenient because they themselves are explicitly a conserved value (total energy), and with their solutions usually being first-order, it's also easier to find their time evolution. Going between L and H formalism is as simple as applying Legendre transform, and they describe the same physics. Sometimes, one is more convenient to work with, that's all. Sometimes one is trivial to solve but problematic to interpret, and vice versa.

I can, and will, go on if provoked.

More to come, I need to check some stuff to avoid long-form redaction later.