Maybe the second law is the wrong way to look at it.

I think so. In practice, changing the surface properties of a body in orbit can affect its temperature. If we coated the Moon with soot it would get hotter, and if we coated it in silver it would get colder.

So our greenhouse layer prevents some power P1 from getting to earth. Earth emits, and P2 is also eaten. The emitted P3 = P1+P2. Earth gets P3/2. The sun is hotter than earth so the power at any given wavelength will be higher, so P1 > P2, therefor P3/2 > P2, which means on net, heat is flowing from the greenhouse layer to earth. However the earth is receiving P1 less from the sun, and P1 > P3/2. So the earth cools down relative to the similar earth without "greenhouse" effect.

Two key complications break this toy model:

1. P1 > P2 doesn't follow from the Sun having higher spectral power. The Sun being hotter just means it emits more power per unit area at its own surface, but our planet intercepts only a tiny fraction of that power.

2. The atmosphere likes to eat Earth's emissions much more than it likes to eat the Sun's. This allows P1 to be less than P2, and in fact it is. P2 > P1 implies P3/2 > P1, which turns the cooling into a warming.

This makes sense to me because the earth is effectively hiding behind a barrier.

The barrier metaphor's a bit dodgy because it suggests a mental picture of a wall that blocks incoming and outgoing radiation equally — or at least it does to me! (This incorrect assumption confused me when I was a kid and trying to figure out how the greenhouse effect worked.)

The assumption that is not true in that model is the atmosphere being in independent thermal equilibrium.

It's a false assumption, but it's not the assumption breaking your (first) model. It's possible to successfully model the greenhouse effect by pretending the atmosphere's a single isothermal layer with its own temperature.

The second model you sketch in your last 4 paragraphs sounds basically right, although the emission/absorption surface is some way below the tropopause. That surface is about 5km high, where the temperature's about -19°C, but the tropopause is 9-17km high. (Also, there's mixing way beyond the top of the troposphere because of turbulence.)