I noticed that there is no documentation to be able to use the Hex Color System in lang files.

When minecraft loads the .lang files it merely loads every string from a .lang file into memory. when minecraft need to draw text it calls the draw function in net.minecraft.client.font.TextRenderer which takes MatrixStack matrices, String text, float x, float y, int color and whereas some functions where we can traditionally change the color such as the chatbox give some leeway. most other screens have hardcoded values for these colors.
Basically, there is no known docs or method to using the Hex Color System in Lang Files.

Was this a forgotten feature? Or was it purposefully ignored / not added

I forgot to mention at the top of the post, but this also goes for other pack files that can use § formatting, like pack.mcmeta, lang files, credits, title screen messages, and other things.