Fully fae means you don't have free will. You are bound to your nature. If they meant for there to be an option to be fully fae and be a PC, then there would be a Fae template.
If the Choice was meant to make them go NPC, then they would be specifically noted (like the Red Court Infected on page 80).
Keep in mind that most of the fey we see Harry dealing with are those in the upper reaches of power: Mab, Maeve, Leanansidhe. That's some serious refresh cost there. And one changeling we saw go fey became the Summer Lady. There's a lot more leeway at the lower levels - when Morgan is ready to kill Harry in
Storm Front for breaking the Fourth Law, the argument is meaningless if Toot doesn't have free will (or something like it).
Last, in the game, the idea of "free will" is a sliding scale. A White Court Virgin in a "Feet in the Water" game who kills using her Emotional Vampire abilities becomes an NPC and loses all of her free will - but the exact same character in a "Chest Deep" game is a viable PC.