Well, if Briscoe actually said that (one can never be sure with newspapers) he is confused.
We believe in continuing revelation from God, yes. However, we do not believe that God speaks only to one prophet at a time.
In fact, anyone who can testify of Jesus Christ is a prophet, for "the testimony of Jesus is the spirit of prophecy" (Rev. 19:10).
We also believe that individuals can receive revelation through the Holy Ghost to guide them through life. That includes guidance to fulfill their callings in the Church. Thus, a bishop can receive revelation to guide the ward over which he presides.
What I think Bishop Briscoe was saying is that God chooses one man at a time to receive revelation to guide the entire Church. That man is the President of the Church, often called "the Prophet" by the members.
.... let us summarize this grand key, these “Fourteen Fundamentals in Following the Prophet,” for our salvation hangs on them.
First: The prophet is the only man who speaks for the Lord in everything.
Second: The living prophet is more vital to us than the standard works.
Third: The living prophet is more important to us than a dead prophet.
Fourth: The prophet will never lead the Church astray.
Fifth: The prophet is not required to have any particular earthly training or credentials to speak on any subject or act on any matter at any time.
Sixth: The prophet does not have to say “Thus saith the Lord” to give us scripture.
Seventh: The prophet tells us what we need to know, not always what we want to know.
Eighth: The prophet is not limited by men’s reasoning.
Ninth: The prophet can receive revelation on any matter, temporal or spiritual.
Tenth: The prophet may be involved in civic matters.
Eleventh: The two groups who have the greatest difficulty in following the prophet are the proud who are learned and the proud who are rich.
Twelfth: The prophet will not necessarily be popular with the world or the worldly.
Thirteenth: The prophet and his counselors make up the First Presidency—the highest quorum in the Church.
Fourteenth: The prophet and the presidency—the living prophet and the First Presidency—follow them and be blessed; reject them and suffer.
~Ezra Taft Benson, Thirteenth President and Prophet of the Church of Jesus Christ of Latter Day Saints.
Does that mean that the Pope is a dual authority with the current president of the Mormon religion? Do we have two religions lifting up Christ, by dual authority, or is there one which is more authorized than the other?