I think that understanding how eyes and light work is very informative on this subject.

Why are there 3 primary colors (regardless of which 3 you pick)? That has nothing to do with the nature of light, and everything to do with the fact that humans see light using 3 distinct frequency response curves [0]. This means that humans perceive color as a 3 dimensional space; and the role of the primary colors is to pick a point in this space by selectively stimulating or masking the 3 response curves. In a world of pure linear algebra, almost any 3 colors would do, but physical reality limits how ideally we can mix them; and how much light they can emit/mask.

Further, the 3 response curves are overlapping, so there is no set of ideal colors that would let you actually control the 3 curves independently.

[0] At least for color perception in a typical human.

Besides having 3 primary kinds of photoreceptors, there are additional complications caused by the initial processing of the color information in the visual system.

Besides the 3 primary color sensations from the red, green & blue receptors, there are a few dedicated detectors for some colors.

Yellow is detected when red is equal to green and both have a high enough brightness. White is detected when red, green & blue are equal and all have a high enough brightness. While black could be detected later, by the lack of information on the other channels, it seems that it also has a dedicated detector.

This gives 6 primary color sensations, where the colors are not perceived as a mixture of colors, like it is the case for orange, blue-green, violet, purple or gray. Presumably these 6 colors correspond to separate outputs of the initial color processing stage of the visual system.

I do not know whether this is true, but I believe that there also exists a dedicated detector for the color brown. While brown is just orange with low brightness, the "brown" sensation is very distinct and very unlike the differences perceived between e.g. dark red and light red, dark green and light green or dark blue and light blue, where the differences are perceived to be only in quantity, not in quality.

A dedicated detector for brown makes sense, because important things in the environment are brown, e.g. the ground is brown in most cases, because it is composed of a mixture of white oxides with red, yellow and black oxides of iron and manganese. Also wood is frequently brown and also many mammals are brown.

Related: some colors can only be perceived by selectively hitting the right cells with tiny lasers.

https://www.scientificamerican.com/article/researchers-disco...

I am surprised that the Purkinje effect and the degree of illumination are not mentioned. For example, should the primary colors be shifted depending on illumination?

https://en.wikipedia.org/wiki/Purkinje_effect

The linear algebra argument for almost any three colors only works if you can have negative intensities. I don't know how to do that with stimulation of photoreceptors, so I don't think that applies here.