I believe that 4300K is pure white on the Kelvin scale.
I'm not sure what you mean.
The temperature of the sun's surface is about 5800K. This is a good black-body radiation source, and corresponds to a peak wavelength of about 500nm, which we see as green light.
I don't think it's any coincidence that our eyes are most sensitive to green, since that is the wavelength given off most by the sun. The other wavelengths given off make up the "white" light.
I assume that 5800K would be the best color temperature for "pure white" light, since that is the color of sunlight. Perhaps a bit lower after the atmosphere has scattered off some of the blue. I measure direct sunlight at 5500K at this time of year.
Granted, we often refer to this as "cool white", and your 4300K example to be something more "neutral white". But that's just terminology. It doesn't mean that 4300K light is more pure than 5800K. In fact, get out that "neutral white" flashlight on a cloudy day and shine it around, and it will look very warm in contrast to the natural light.
Our eyes are good at adapting. Personally, I find anything from 3500K to 6500K to look pretty neutral and "pure white", when it's the only light source. I don't really start seeing blue until it gets above 6500K. Past 7500K, and it starts to look pretty ugly. Although, I'm not sure if it looks ugly because of the temperature of the light, or the horrible CRI at that point.