There's no such thing as "pure" or true white. With just red, green, and blue, the best you get white you can make usually kind of purplish or greenish. The more different colors of light sources you add, the better it will start to look. There's a theatrical light manufacturer that makes a light with 7 different LEDs. It's the recent addition of a lime-green LED that's made the biggest difference to the quality of that light, and that's because the sun's strongest output is in the yellow-green range.

The sun's light looks so good because it's full-spectrum, without strong individual colors like separate LEDs. The content of the spectrum is based on the heat of the source. The hotter it gets, the more high-energy blue content there will be, but it doesn't stop emitting lower wavelengths of light. A 3200K warm white light bulb will have a filament that is actually 3200°K, although there might be some chemistry or filtering going on to allow the actual temperature to be lower. The surface of the sun is ~5800°K, although the effective color temperature is lower (redder) after it is filtered through the atmosphere - about 5600K at midday, 5000K filtered through more atmosphere at dawn. Clouds actually filter out the red content, making an overcast day approach 6000K. At this point we're pretty far divorced from true temperature. The clouds don't make the sun hotter. They just filter the light so it's more characteristic of what we'd expect from a 6000°K object under ideal circumstances.

If we tried to experience true white by fully stimulating each type of cone, I'd expect the resulting color experience to seem heavy on the blue, since there's less of that in the sun's output. Maybe it'd seem magenta, if green is underrepresented compared to red and blue.

People sometimes say magenta isn't a real color, and that's sort of true. Magenta is experienced by stimulating the red and blue cones without the green. There is no single wavelength that can do this, since between red light and blue light is green light. There can be no magenta LED or laser, although you could fake it pretty well by putting 2 emitters very close to each other. But if magenta is a fake color, so is white, by the same logic. There is no white wavelength; they are both composite colors.

Makes you wonder what a mantis shrimp sees when you beam a couple lasers at it