Actually, The Sun outputs much more than these six wavelengths. The figure provided shows the spectrum of the Sun:
As you can see, the Sun outputs light along a continuous curve at all visible wavelengths. The combination of which appears white to our eyes as a byproduct of having evolved in orbit of this star.
The image also shows the absorption effects of the atmosphere. If you specifically discuss the blue scattering from the atmosphere, the following figure is a nice depiction.
As you can see, the blue scattering leaves only the more red visible wavelengths, which makes sunlight appear more yellow around midday and more red near dawn/dusk.
As for why we can simulate white light perfectly without having to reproduce the solar spectrum; that has everything to do with how humans perceive colour. The human eye has special cone cells capable of perceiving colour. As shown below, there are only three true colours we can see; red, gree, and blue (with more green cells than the others again due to the solar spectrum).
It is by using a combination of these three colours that we interpret all possible visible wavelengths. Our brains interpret the relative strengths by which each different colour cone is stimulated and assigns a different colour to each combination. The next figure shows the sensitivity ranges of these cones.
The three types of cones are sensitive over all visible wavelengths but are stimulated differently by each wavelength. Thus, we can interpret a nice teal when a 500nm wave stimulates the cones appropriately, or we can simulate this for human eyes (and this is what we do with things like computer screens) by combining certain levels of 400, 535, and 700nm (thanks to Peter Shor for the correction) lights to stimulate the cones by the same ratio as the 500nm light. Being the same ratio, our brain is tricked into seeing it as teal, but the summed light itself is not actually teal. In the same way, a combination of the six colours you mentioned would look white to us, but to make a true white, the Sun must output across all wavelengths in the visible spectrum at varying amounts.