What does the combustion chamber actually do? It's a place for the
propellants to mix and burn subsonicly. The mixing typically requires
ferocious turbulence.

The problem I have with mixing and burning simultaneously is that a
lot of the mixing is happening at high temperatures, where the gases
have large volumes and require an enormous amount of turbulence to
mix. My guess is that this turbulence is a way to lose energy. An
order of magnitude less turbulence is required to mix the propellants
before burning.

Suppose the propellants are not hypergolic. Why not mix them, then
burn them, more like a bunsen burner? My understanding is generally
that any gas acceleration that happens while heat is added to the gas
(example: bunsen burner) increases entropy, but that acceleration that
happens with no heat added (example, rocket nozzle) does not. This
means you want the burning to happen at the lowest gas velocities
possible.

Deflagration in premixed gas phase propellants is pretty slow. It
gets faster as the initial pressure increases, and as the initial
temperature increases. I'm going to guess that methane-oxygen is
around 16 m/s at 10 atm. The burned gas velocity is going to be
around 160 m/s. Detonation pressures for methane are 20-40 atm, which
is why these operating pressures seem low.

To capture the flame, I'd use a tube that flares, perhaps with wires
crossing it to form a flameholder (and to provide initial ignition).
The flaring section needs have a large enough area ratio that the
flame is still trapped for any expected variation in flame speed due
to startup temperature and pressure transients.

After the flaring tube, I'd have the flow reconverge in a de Laval
nozzle. So the engine ends up looking basically like all rocket
engines do, except instead of a flat injector plate I have a flaring
tube, and the propellants injected at the end of that tube are
injected gas-phase. I might have other (not hot) wire grids in the
throat to increase the low-temperature turbulence and mixing. The
area ratios of injector throat to combustion chamber to nozzle throat
might be 1:10:1.

Now consider start-up. Begin running 100% fuel (gas phase somehow) at
full operational mass rate. Run current through the ignition wires to
get them hot, then add oxygen progressively until you get to operating
mixture ratio. The flame starts at the wide end of the flare, then moves
down near the throat as chamber pressure builds.

Advantages:
- Less noisy
- Potentially better Isp across smaller chamber/ambient pressure
drops, from lower turbulence
- More complete burning
- Short "combustion" chamber
- More laminar flow near walls means
- less heat transport to walls
- film cooling works better, with less wasted fuel
- less scouring action
- rocket engine scales down better (same Isp at smaller size)
- Reliable, nonexplosive startup
Disadvantages:
- Extra drag from larger nozzle convergence ratio
- Need to boil propellants before injection
- More constraints on cooling system (film cooling less usable)
- Smaller input/output temperature ratio will lower Isp
- Larger effect at higher pressures

How thermodynamically efficient is accelerating a boiling two-phase
mixture? My guess is that boiling the propellant in the combustion
chamber is not critical to efficiency, since the SSME injects
gas-phase hydrogen. Perhaps higher-temperature methane, propane, and
oxygen are a different story.

If you haven't figured it out already, this idea is targetted to a
restartable upper-stage engine that is simple enough for amateur work.