> [The surface temperature of the moon is] an absolute limit on the amount of incoming irradiance you can create to your object.
It's the radiance (brightness) of the moon (as perceived at the moon) that is an absolute limit on the incoming irradiance you can create, because of conservation of etendue.
Separately, the fact that moon rocks (which experience that exact amount of irradiance) reach some given temperature X°C while losing very little heat to conduction (certainly less than an object on earth would), shows that this level of irradiance is insufficient to heat up your kindling above X°C.
This argument has nothing to do with the moon being a black body, or an approximate black body, or any such thing. Just the fact that moon rocks, which are exposed to this light (with very little heat conductive losses), function as a kind of a thermometer which tells you how hot that light can make something[1]; and the answer is "a bit over 100°C"
> Since the moon absorbs about 90% of the light incident on it, we can assume that the surface temperature is lower than it would be if it was a perfect black body, presumably reaching a temperature corresponding to a sun that was about 10% less strong.
No... An object with 90% absorbance absorbs 10% less light energy, yes. But it also emits 10% less light energy, so the two effects cancel out and it reaches the same equilibrium temperature as a black body.
[1] https://en.wikipedia.org/wiki/Effective_temperature