Roughly, if the average color of something is white your eyes will make it look white. Since we only have three color receptors you only need red green and blue in the proper amounts. The rest of the visible spectrum is naturally interpolated.

An approximation of white doesn't work for reflections which is the main reason this method isn't ideal. Real world objects can reflect different wavelengths very unevenly so even if these sources appear white directly their reflections take on a variety of strange hues.

The phospor spreads out the peaks so the light spectrum emitted is smoother, closer to black body radiation which is the type of light emitted by the sun. Humans are wired for sunlight so this makes things look better.