So, in fact, if humans had more than just 3 kinds of cones, we could see a much wider range of colors.
There are a few related but separate ideas here.
There are internal subjective colors, the qualia, and then there are the signal generated by the eye. More sensitive eyes don't make a "wider range" of subjective colors in the brain. The brain still needs to map the signals to subjective colors. This explains why we don't just evolve infra-red sight or something, since we'd have to map something like red onto that, and then we couldn't map that onto the existing spectrum anymore, meaning we're stretching less colors to cover more of the spectrum. This is probably the best way to understand things. Colors are in the brain, then signals from the outside world are mapped to the brain's color-space however they can fit. It's actually an illusion to believe that color resides in the light itself or in the cones of the eye, even. The cone only either send or don't send a signal, they don't send "color".
You have the people with tetrachromacy, a 4th type of cone, and some of them can definitely tell more colors apart. However that's almost certainly them having finer gradations between existing colors since they have more variation in eye-signal to work with. It's as if everyone else is working in 8 bit color and they're working at 10 bit color. More points between existing colors, but not a wider "range" of colors.
The real interesting thing however would be asking about how the qualia work. For that, the primary color theory is actually the most natural. It's more intuitive to see how red blue and yellow are unique and that mixing these makes the other colors. The mechanics of how rods and cones work in eye is just an implementation detail of the sensors, and things like red+green light mixing to yellow is just an artifact of how that's mapped to the qualia in the brain.
So in other words trying to say anything
fundamental about the nature of color by studying the properties of light mixing is incorrect, since it's circular logic. Yellow light is just a specific wavelength of light, and it's not "made of" green and red light. We can fool ourselves otherwise by mixing red and green light and apparently, but not really, getting "yellow" light. Similarly, red + green + blue doesn't really "make" white. White and Black were probably the original qualia, representing light vs not-light, and as the vision system evolved, the system retained "backwards compatibility" by saying that if all receptors are active, that remains as "white".
As for the actual qualia, the in the brain stuff, my hunch is that the primary color system is the closest to the subjective reality and there are 5 basic color qualia: white, black, red, yellow, blue, with the others being mixes of these qualia. And the real fun question here is whether other qualia are possible, and how would be know? For example if you had the same system minus the blues, then you'd see the whole world in shades of red, orange yellow, and you couldn't conceive of blue, green, purple colors at all. We'd have the same issue imagining an unknown color component, but it's definitely possible even with our existing brain design.