The type of mosaic which tyou are talking about is RGBW. Telling about why it is not used is complex and I may misinform you. I may point at least one thing which in my opinion is most important.
So, my question is, what is the main reason we don't see products containing the alternative color filter described above?
Every channel of output image is constructed from all three channels of sensor output - they are mixed using matrix conversion. The farther the spectral response of a sensor is from LMS response the bigger is the error which that conversion introduces. White is kind of worst scenario of this. Suppose that you succeeded in making
W channel which equals
R+G+B. Then, to get each component from that channel using adjacent values you will need to compute
R=W-G-B and so on. This will introduce very big errors and will probably cancel all the benefit from having
W channel at all. Same applies to CMYK sensors: they require very strong conversion coeffificents which introduce the colour errors.
The CMYK and RGBW cameras will produce better mono-converted image for the same amount of light but RGB sensors will produce better RGB image even though their quantum efficiency is like half of CMYK sensors - it just happens so because of noise sources.
My guess is that optimum sensor mosaic (i,e, producing smallest noise for a given amount of light) would be 1:1:1 RGB instead of 1:2:1 RGB and the responses would be close to LMS response but I cannot backup this statement ATM.
A downside of this method is that roughly 2/3 of the light is not detected.(exact amount depending on the light and configuration of the filter)
Due to overlap between the responses of the Bayer filter dyes it's closer to 50% light loss (one stop) than two thirds.
Both statements are a far cry from real world. Canon 5D Mark II looses from 2/3 to 4/5 of light at the peaks of sensitivities and, when compared to ideal mono (unmasked) sensors, it looses a whoopping 9/10 of all light (i.e. the mono image from Bayer camera will have approximately same noise as the image from mono camera made from 1/10 of the light).
Check out the scale on the left of the graph:
More modern cameras may be better at it but I do not have the data to compare. There are causes for light loss othen than mosaic pattern.