Well we do see the day lit part of the moon if we actually look. If we knew where to look we'd see some brighter planets and stars as well. See 10 surprising space objects to see in the daytime sky and also read through Naked-Eye Venus Apparitions, Conjunctions & Elongations 2010 to 2020.
Raleigh scattering of visible light in Earth's atmosphere is wide angle. Photons that get kicked out of their initial direction go all over the place
If you are looking at the moon, most of the photons from the moon traveling towards your eye make it there un-scattered, so no definition is lost, and the ones that are scattered go in very different directions, so the moon is lightly dimmer but just as clear and crisp as it would be at night. However the scattered light from the Sun brightens the scene, so the dimly lit part of the moon (lit only by Earthshine) is not easily noticed.
If you put a large black spot on the Earth and viewed it from LEO, it would appear sky blue to you as well! However because the reflected light from the Earth is so bright it would be hard for you to see it, your eyes would be overwhelmed by the bright reflected light from the Earth.
However if you used a straw or hole in a piece of paper or whatever you had available on the ISS and blocked the excess light out, you'd see the blue sky above the (now mysterious) large back spot easily.
above: Venus and the Moon during the day, from Earthsky.org, photo cretid nakedeyeplanets.com.
above: Venus seen through a telescope during the day, from here.
above: The Moon seen through a telescope during the day, from here.