Why does mixing every paint colour produce gray instead of white?

You are confusing additive and subtractive colour mixing. If you mix paints together you should get black, not white.

In additive mixing (as used in TVs and monitors), you create light, which is then mixed. When you mix the three primary colours (red, green and blue), you produce white. Other mixes produce other colours, for example red and green combine to produce yellow.

When you use paints, you are using an external light source (the sun or a light bulb) and each paint reflects some of the wavelengths and absorbs others. For example, yellow paint absorbs the blue wavelengths, leaving red and green, which mix to yellow. This is called subtractive mixing, and the primaries are cyan, magenta and yellow; when you mix paints of these colours, the result is black. Adding additional colours to this mix keeps the result black, as there is no more light to reflect. Other colours are made up by mixing the primaries.

With both additive and subtractive mixing, the result of mixing colours depends on the purity of the primaries. No paints are "perfect" cyan, magenta or yellow, and as a result the mix will not be completely black. You may get a dark brown or purple, depending on the paints you use. This is one (of several) reasons why printers use black as well as CMY.

The same goes for monitors: you never get "pure white" - which is typically defined as light with a colour temperature of 5500K, about the same as sunlight. Some monitors can be set for different temperatures. Some are set to 9000K, giving white a bluish cast. Interestingly, the colours that can be displayed on a monitor do not match those of a printer (or paint). A monitor can display colours that a printer cannot print, and vice versa. Every device has its own colour gamut, usually smaller than the eye's gamut, so with any device there are colours we can see but which the device cannot produce.

The reason why all this mixing occurs is because our retina has sensors for red, green and blue, and the brain mixes these inputs to tell us what colour we are seeing. This is why the primaries are RGB, or CMY.