As a matter of fact, this has been submitted to direct experiment by zetetic scientists, and it has been demonstrated conclusively (besides being common knowledge anyway) that the Moon's light is not attended by even the slightest heat. Whither has the Sun's heat gone, if, as you claim, it is reflected onto the Moon? For as you rightly assert, the Sun's light ought to be traceable by virtue of its temperature characteristics!
You are correct that the Moon's light does not contain a lot of heat, even though it is spectrally identical to the Sun's light. How is this possible?
First of all you should understand that information about the temperature of the source is contained in the colors of the light, or the spectrum. It is much like hot iron, which changes color from red to white to blue as it is heated up. Just by observing the color one can estimate it's temperature quite accurately.
And second; the amount of heat that is actually transported in the light depends on both the colors
and the flux, or intensity of the light.
So if Moon light is indeed (diffusely) reflected light from the Sun, it means that it must be diffusely reflected in all directions of the sky and that the Earth consequently receives only a small fraction of this light. Which automatically means that the intensity has significantly dropped compared to direct sunlight. Even though the colors haven't changed. In other words, we see through the colors/spectrum that Moon light is produced by a hot source (exactly as hot as the Sun) but it because it is diffusely reflected it has lost the original intensity and doe therefore not carry a lot of heat.
I am sitting in a room with white (diffusely reflecting) walls, illuminated by a few light bulbs. The color of the light coming from the walls is identical to the light of the bulbs, but it does not carry significant heat (as the light bulb itself does). It's the same...
I hope this makes my point clear.