This is a great exercise, especially when you limit yourself to using only the channel mixer. This would include setting your brightness, which makes this (a lot?) harder to do due to this being an RGB model tool where brightness and saturation are linked (unlike the L*a*b model, where luminance and chrominance are fully separated).
I find this part to be true, although comfort level using a specific tool does come into play.
@Underexposed
If you let go of the R=G=B needs to be a specific number requirement and fall back to the R=G=B need to be the same while inside the channel mixer, 'cause you change brightness/luminosity much easier/better using other tools, this becomes easier to do.
The actual R, G and B numbers become a guide to which colour needs adjusting. Starting with the most dominant colour (orange-> RED in the cat example) and just using the major channels (red in R, blue in B), you can initially ogle it. For the fine-tuning you need to use the secondary channels and now the numbers are needed to get to R=G=B.
I personally try to stay away from the G channel, if at all possible. I see this as an anchor and noticed that changing these have a big(ger) effect on the R and B channels. This isn’t always possible though, but keep the changes as small as possible.
You could ditch the channel mixer altogether and try doing this with the RGB curve tool using independent channels and the help of its picker