I think everyone's assuming the dye will be complete but I think it'll follow the examples we've already seen. Some wings and tail feathers may be coloured but I think the bulk of the birds will still be yellow. Take for example the black tipped legacy chocobo or the white tipped dual rider chocobo for recruiters. It seems like that's the extent of the unique colouring: tips only.

If full body feather dying becomes a thing, that'd be great. I just wouldn't hold my breath for it after a couple of examples against that concept.