The Patriarchal office is by lineage or descent. That way it cannot be stolen by an interloper; thereby creating a separation of power inside the one Church (or kingdom).
Joseph became President through divine ordination by the Lord and messengers sent by the Lord.
Brigham Young was sustained as President, relying upon his ordination as an Apostle.
John Taylor was also sustained, relying also upon his ordination as an Apostle.
These precedents were relied upon through Joseph F. Smith, who had an ordinance/ordination accompany his assumption of the office of President of the Church. That ordination was performed by his half-brother, John Smith, the Patriarch of the Church.
Heber J. Grant was conflicted about the Patriarch because he considered himself a descendant of Joseph Smith by sealing and the Patriarch was competition to that; and therefore he did not want the Patriarch to ordain him president. He had the Twelve ordain him. He also initiated the name change from “Presiding Patriarch” to “Patriarch to the Church.”
Interestingly the term “Prophet” was not applied to a living man holding the office of “President of the Church” until 1955, during the administration of David O. McKay. The term “Prophet” until that time always meant exclusively Joseph Smith, and not the office holder of President. Before then it was “President Young” and “President Taylor” and “President Woodruff” and so on. However, in 1955 the Church News began a new practice of referring to the living President McKay as a “Prophet.” It was felt that changing the reference to the living President would result in quicker acceptance of direction from him, and less criticism of the President. (President Grant was the most unpopular Church President in the Church’s history, and that was something they hoped to avoid happening again.) It worked. No-one wants to reject counsel from a living prophet of God.
So since that time the practice has been for living Presidents to continue to be referred to by the title “Prophet” by all General Authorities and other leaders. However, I have noticed that the President never refers to himself as “Prophet” in any declaration I have been able to find. He accepts that term as used by others, but does not apply it to himself.
The recorded times when a Church President was asked if he was “a Prophet” include testimony by Joseph F. Smith when asked by the Senate Committee in the hearings to seat Senator Smoot. His response was “my people sustain me as such.” President McKay was asked by a reporter and his response was “look me in the eye and tell me I’m not a prophet.” President Lee essentially repeated the same response to a reporter as President McKay. And when he was interviewed by the Press President Hinckley essentially repeated Joseph F. Smith’s response, saying in effect: “I’m sustained by the Church as such.” There may be others, but those are the ones I recall at the moment.