One reading of these verses would be that having a king leads people away from God. Even though that is a simple reading of the text, the social and cultural background provides the complexity to understand better what was happening.
Foremost, it cannot be a condemnation of kingship specifically, because the Nephites have had a king for much longer than they have not at this point. Thus, the answer is not in kingship, but in some other problem associated with kingship. When the Nephite nation was first created, the people wanted Nephi to be a king, but Nephi himself was reluctant. Where had that desire for a king come from?
The Nephites lived in a larger cultural area, subject to the influence of the people around them. Early on, it was a time when the establishment of kings was becoming the most common political system, and therefore the Nephite people desired to be more like their neighbors. That outside influence often affected them, typically with the temptation toward social hierarchies, at the top of which was a king.
Thus, what is happening in the land of Zarahemla is that the old influences of the people who had been in Zarahemla prior to the arrival of the Nephites were returning to their old beliefs and reflecting, perhaps as they once had, the cultures around them. It was those ideas that were detrimental to the church, because they espoused different religious ideals. In particular, they denied the coming Messiah, which was a foundational element of Nephite religion. More than just desiring a king, these men who desired power also desired the destruction of the uniquely Nephite religion and culture. It was the potential destruction of their religion that caused the need for the missionary journey of Helaman and his brethren.