I can see that point of view, but the earlier you claim it, the better overall it is for town. With the Innocent Child, there's zero reason to claim it early since all you need to do is just reveal yourself publicly when it comes down to it and confirm yourself.
Miller and Beloved Princess are two roles that generally need to be claimed early otherwise they cause issues just like this.
It also heavily depends on if the beloved princess skips day phase through death period or if it's conditional, and as you've already pointed out. A total death Beloved Princess is a role that nobody wants out in the open because it causes all sorts of issues regardless for town. It's the definition of a negative utility for town
