It is being led by a prophet makes it the true church. It is being led by a prophet, by a seer, by a revelator that makes this the true church of Jesus Christ.
When the church was organized in April of 1830 what did it have? The members had the Book of Mormon. Joseph Smith and Oliver Cowdery had been given the priesthood. Baptism and the gift of the Holy Ghost had been restored.
It had a revelator at its head. But otherwise, it had only a fraction of what we have today. No temples. No sealings. No redemption for the dead. No knowledge of three degrees of glory.
But it was still the true church.
It was a living church. It was a living church, because God stood at the head of it. That is only possible when a prophet stands at the head, because God stands revealed, or remains forever unknown.
However, the priesthood is useless as a method of government without a revelator at the head. Without the head, it is an army without a general, and swiftly falls. This is fundamentally not a grassroots church. The day it becomes one is the day it ceases to be the true church, for it is the fact that we are led by the prophet of God that makes it the true church.
When Lehi went into the wilderness, did he have the true church with him, even among such a handful of souls? Yes, for Lehi was a prophet of God, and therefore God stood at the head.
As long as God stands at the head, then it is God's church. But God only stands at the head when a prophet stands at the head, because no man can know God except by revelation.
Today it is Thomas Monson that is God's prophet. He stands at the head of the church. He is the only man authorized to hold all the keys of the priesthood, and the only man who can receive revelation for God's church.
We are led by prophets, by seers, by revelators. They are the first presidency and the quorum of the twelve apostles. And that is what makes it God's true and living church. We gather souls to Christ by gathering them to these men, where those we gather can come and hear the pleasing word of God.