The moon does significantly redden when it's close to the horizon, especially if you can see it over the ocean or from very flat ground, where you have a clear view all the way to the horizon. But even at best, it's more an orangening. Why is the moon's reddening less obvious than the sun's?

I think it's important to think about what 'reddening' means. You said "the spectrum of moonlight is more redshifted than that of the sun, which should contribute to an even more intense reddening" (I would not use the term "red shift" which has a specific astronomical meaning) -- but that's actually the opposite of the case.

The sun at the horizon turns red because the blue light is being refracted around the atmosphere (contributing to blue skies for the day side of the world), and the red passes straight through. Consider if the sun produced only red light -- then you would see no additional reddening near the horizon because there's no blue light to remove from it.

So if the moon's spectrum is already redder than the sun's, then we would expect its reddening to be *less* intense than the sun's, not *more* -- there's not as much blue light to remove, so the change is smaller.