In fact, I've always felt that white mint chocolate chip was more premium, but it's definitely gone out of style a bit. I also believe it tasted slightly different - at least the bryers, it was maybe more of a spearmint?
To top it off, the issue isn’t dyes, it’s artificial dyes. You can still dye your mint ice cream green, it will just be a slightly lighter shade of green.
As for what flavor would warrant dyes? Off the top of my head: Mint, peppermint, strawberry, orange sherbert, blue moon... I'm sure there's more.