Why do complementary colours desaturate each other?
Just to get some terms defined:
Saturation:

Color saturation refers to the intensity of color. When color is fully saturated, the color is considered in purest (truest) version. Techopedia.com

Complementary color

a color that combined with a given color makes white or black OED

In painting, here's why a color's complementary color reduces intensity (desaturate)

First, each of the six pairs of Complementary Colors always contain one cool and one warm Hue. This means they are the colors with the most contrast.
Each pair of Complementary Colors always contain some combination of all three Primary Colors. For example, in the pair Yellow and Purple, Yellow is one Primary and Purple is a mixture of the other two Primary colors Red and Blue.
When you mix all three Primary Paint Colors together you get a dark neutral. So because Complementary Colors always contain all three Primary colors,  they quickly neutralize each other when mixed together.

From Color-Wheel-Artist.com There are some very good illustrations at the link.
Bottom line: When you mix complementary colors you are blending all three primary colors which reduces intensity/saturation.