> That doesn’t alter the point that he’s still responsible for the things happening in his name.

There's case law on this matter actually, coming from the trial of King Charles I.

> The court challenged the doctrine of sovereign immunity and proposed that "the King of England was not a person, but an office whose every occupant was entrusted with a limited power to govern 'by and according to the laws of the land and not otherwise'."

So no, granting the Royal Assent doesn't happen in the name of King Charles III.