What is the benefit of invoking blessings upon the Prophet (peace be upon him), his family, and Abraham (peace be upon him) after their passing, and does it entail sin or a doctrinal error to suffice with only sending salutations and peace without invoking blessings?

The Ibrahimi prayer encompasses the life of the Prophet, peace and blessings be upon him, and after his death. There is no issue in supplicating for blessings upon him, as it is a prayer for an increase in the good that Allah has bestowed upon him, for its continuation, and for its spread. Blessings after his death include the multitude of his followers, the spread of his call, the elevation of his status, and the rewards for those who act upon his teachings. Ceasing to supplicate for blessings upon the Prophet, peace and blessings be upon him, in the Tashahhud is a mistake and a lack of etiquette, for the Prophet, peace and blessings be upon him, taught us the Tashahhud which includes supplication for blessings upon him.