Why do pure colors (red/green/blue) become a mixture of colors when converting raw?

The short answer is that the "red", "green", and "blue" filters in your camera's Bayer mask are not the same colors as the Red, Green, and Blue colors used by emissive RGB displays. Neither are the "red", "green", and "blue" colors to which the three types of cones in our retinas are most sensitive.

Here are the sensitivities of the Short wavelength, Medium wavelength, and Long wavelength cones in our retinas, with the curves for each the color that we perceive for the wavelength of that type's peak sensitivity.

Typical Bayer masked sensors are similar, though the "red" filter is a little more "yellow-orange" than "yellow-green".

Here's the same graph with vertical lines drawn where a typical RGB display (or RYGB display, which adds a yellow channel) emit their light. Notice how much distance there is between the peak of each "red", "green", and "blue" channel on a camera's sensor and the Red, Green, and Blue channels emitted by an RGB display. Particularly, please notice how much closer the "red" filters in our Bayer masks are to yellow than to red.

The lack of true Red in the Bayer mask also helps to explain why sensors are green.

All cute little drawings on the internet notwithstanding, the "red" filters in most Bayer masks are centered at around 590nm, which we perceive as an orangish shade of yellow, and not red at around 640nm. There are also more subtle differences between the "blue" and "green" filters and the colors used by RGB displays.

So the "pure" Red color emitted by an RGB display at about 640nm creates a response in more than just the photosites masked by a filter most sensitive to 590nm but with significant sensitivity all the way from 560nm to 790nm or so. The "green" filtered photosites also respond to 640nm Red light. Everything past 790-800nm is filtered by the IR cut filter in the filter stack in front of the sensor (which isn't placed in front of the sensor when sensitivity is measured).

Likewise, the "pure" Green color emitted by an RGB display at about 530nm creates a response in more than just the photosites masked by a "green" filter. The "blue" filtered photosites also register a response. Ditto for the 480nm light emitted by the display's Blue channel. Both the "blue" and "green" filtered photosites on the camera sensor register a response to that light.

We make our cameras this way to emulate the way our eye/brain system creates the perception of color from certain wavelengths of electromagnetic radiation. The only reason we call a portion of the electromagnetic spectrum visible light is because that portion of the EMG spectrum creates a biological response when it falls on the cones in our eyes' retinas. There are no specific colors implicit in certain wavelengths of light, there is only the perception of color created by the eyes and brain that perceives it. Animals with cones that have differing responses to the same wavelengths of light do not see the same colors for the same wavelengths and combinations of wavelengths.

For a camera to create "pure" colors when pointed at an RGB display, one would need a sensor that has a "red" channel that does not respond at all to Green or Blue light emitted by the RGB display, a "green channel" that does not respond at all to Red or Blue light, and a "blue" channel that does not respond at all to Green or Red light. But such a camera would not be able to construct any colors other than pure Red, pure Green, and pure Blue. There would be no way to synthesize other colors using overlapping sensitivities of "red", "green" and "blue" filtered photosites that mimic the way our retinal cones and our brains combine to synthesize colors based on the overlapping sensitivities of our S, L, and M cones.