My understanding the concept of the 5 piece French wardrobe is that you only add 5 pieces each season or year or whatever. This apparently requires one to have a wardrobe that is complete, with all the needs covered; the 5 pieces can then be anything you feel appropriate for you. So, for me, I would probably add 5 sweaters this winter, since that is what I would get the most use out of; and I would likely add whatever colors struck my fancy.
I think most of those capsule or French wardrobes use those basic colors because they mix and match with everything. The idea is to get versatile pieces that will be fashionable or stylish for a long time. Going back to my sweaters, I would likely add zero black sweaters because I have too many already; I would add a soft purple, a screaming cobalt, a rich chocolate, a soft blue, and a blush/soft pink (Angie's love of whites and creams is rubbing off on me LOL). For me, those would be versatile colors that would go with mostly anything in my closet.
If you're doing the 5 piece wardrobe, add what works for YOU. If that happens to be emerald, gold, purple, orange, and chartreuse, so be it.