I'm not sure why. If people have a legitimate and earnest Religious Exemption, why does getting a statement from a Priest increase the chances of the approval?
Because it should not be a “religious” exemption. It should be a “conscience” objection. A person can be an atheist, agnostic, or a member of any religion and still conclude the vax is morally problematic. The Church’s catechism states vaccines must never be forced or coerced against a person’s individual conscience.