To what extent is it correct that the saying: "May Allah's salat and salam be upon him and his family" is more correct than the saying: "May Allah's salat and salam be upon him"?

It is permissible to invoke blessings upon the Prophet (peace be upon him) without mentioning his family, based on the Almighty's saying: "Indeed, Allah confers blessing upon the Prophet, and His angels [ask Him to do so]. O you who have believed, ask [Allah to confer] blessing upon him and greet him with [explicit] salutation." However, it is better to invoke blessings upon him along with his family, as in the Ibrahimiyya prayer: "O Allah, bless Muhammad and the family of Muhammad, as You blessed the family of Ibrahim. Indeed, You are Praiseworthy and Glorious. O Allah, grant mercy to Muhammad and the family of Muhammad, as You granted mercy to the family of Ibrahim. Indeed, You are Praiseworthy and Glorious."