I think one thing which would help me develop an intuition for it would be to see the calculation of the lens size for heating a one square-centimeter area on the earth to as high a temperature as possible by the light of the moon, and what that optimal temperature is.
Anyone reading for whom this is straightforward? Even a description of how to do the calculation would go a long way.
For a single lens you just want something really big. Make it take up 90+% of the sky from your target.
For temperature, the no-calculation way is to measure a rock on the moon (article says 100C) and use that number for how hot you should be able to get.
The calculation way goes as follows: Near earth you get 1400 watts per square meter of sunlight, so if that bounces perfectly off the center of a full moon and gets through the atmosphere with no losses, your target will get 1400 watts per square meter. That's equal to a black body at 122C. After taking into account the spherical shape and atmospheric losses you might get less than half of that, so ambient heat might drown out your results.
You can make an enormous lens where all the energy coming off a particular acre of moon goes through it. But you can't focus all of it onto a spot smaller than an acre, no matter what you do.
The current top comment here seems to agree with my original intuition, though.
This is the optical argument for maximum concentration given conservation of étendue (the same page has an optical argument for the conservation):
https://en.wikipedia.org/wiki/Etendue#Maximum_concentration
The angle subtended by the moon is approximately 0.54 degrees, or about 0.01 radians, so the maximum concentration factor is about 10,000. The moon provides about 0.1 lux of illumination, so the maximum illumination you can achieve by optics is about 1000 lux. The sun subtends about the same angle, and we receive 30,000-100,000 lux from it, and the maximum illumination you can achieve from concentrating it optically is about 1B lux.
I'm willing to believe that you can't light a fire from the moon, if the intensity's a million times lower.