The light from the moon is not seen because the moon is not lit. The moon is not another sun that is just not as bright. The moon is a created light that has a different function than the sun. The Bible never states they have the same functions or that the moon light is to be a continuous light like the sun.
Then God said, “Let there be lights in the firmament of the heavens to divide the day from the night; and let them be for signs and seasons, and for days and years; and let them be for lights in the firmament of the heavens to give light on the earth”; and it was so. Then God made two great lights: the greater light to rule the day, and the lesser light to rule the night. He made the stars also.
The Bible only states the moon is to rule the night. The sun has never been seen out at night, so the moon is the light for the night, and when the moon is out during the day. The sun's light is greater than the moons light, so the sun's light rules the day even when the moon is out.