Perhaps the people should listen to Amulek due to his lineage or wealth, but Amulek also understood that such circumstances might not allow everyone to relate to him, even should they listen. Therefore, he tells them the important information that they could relate to. He is like they are.
Even though Alma had declared that the people of Ammonihah had received the gospel teachings, Amulek personally knew that not all who had the opportunity to hear would have heeded what they heard. He was one of those who had not been very interested in “the ways of the Lord.” Modern readers might see this as a declaration that he hadn’t been a religious person, although we should remember that our modern perception doesn’t replicate the ancient situation.
Even more importantly, Amulek says that “I was called many times and I would not hear.” This places him in the position of standing for all of the people of Ammonihah. Alma had accused them of being called, but not hearing. Thus, Amulek puts himself forward as the model of such a person of the city.
Alma had promised that the people of the city could repent. Amulek models that repentance. He tells his story, and places it firmly in a time that everyone could understand. It was recent. If we can accept the accuracy of the statement that it was on the “fourth day of this seventh month,” it would appear that the seventh month had not yet ended when Amulek told this story.