First off - you are talking about two different things. The 6x6x6 color cube does not have color planes with constant intensity. Each color plane just locks one color channel while ramping (spanning) the other two color planes.

The formula I posted isn't there to adjust white. It's there to compensate for the spectral response of our eyes - but normally hidden by the candela definition for the diodes.

Note that there are many "white" in this world. You change the balance between warm or cold white by adjusting the balance between red and blue. Which color temperature you want depends on what color temperature you have for other illumination. Having a warm white sign lit by a noon sun will make the sign look very red. Having a high color temperature when lit by a light bulb or a sunset will make the sign look very blue.

The equation 0.3*R + 0.59*G + 0.11*B is not used to create "white". It is the constants used when converting an RGB image to grayscale, to make the different colors of the original image give a similar darkness/contrast as the original color image. It is an indication to how our eyes has different sensitivity to different wavelengths.

See this graph:
http://www.amastro2.org/at/ot/othcs.html

Since the sensitivity varies, you can't take 100mW of green light and 100mW of blue light and saying that the two have the same intensity. The amount of power emitted is the same, but our eyes considers the green light much stronger - we are almost 6 times more sensitive to green light than blue light. This means that R+G+B = constant does not work when modulating power to get constant intensity. You need 0.3R + 0.59G + 0.11B = k.

This difference is color sensitivity is then camouflaged by the "candela" definition, which takes into account the eyes spectral sensitivity. This means that a 10 candela red LED emits about twice as much power than a 10 candela green LED, but we will perceive them as similarly strong.