You're using the past tense here, which is unwarranted. The Golden Stool remains central to the ritual and spiritual authority of the Asantehene, Osei Tutu II, today. The idea is that this particular stool was called down from heaven for Osei Tutu I by his spiritual advisor, as a symbol of rulership at the time of the uniting of the Asante Confederacy after the defeat of Kumase's rival Denkyira (ca. 1702). The reason it's important has to do with cultural norms and the scaling up of authority from the household to the wider nation.

Most Akan households (and those of many other societies in the region) have stools--objects handed down, usually with ritual value, sometimes leaned against walls for guests, sometimes symbolically there for ancestors. The stool is a sort of repository for the soul of its owner, and those of rulers embody some of their authority, which means that the Golden Stool is a repository for the soul of the nation by that logic. It is gold in signification of the wealth and power of the people who live atop the goldfields, but also to reinforce the hierarchy of stools. The Asantehene never sits on the stool, but is lowered and raised over it. In that sense, it is the royal stool, but it's not a royal's own. It's part of a wider refashioning of national unity, identity, and ritual that helped to create a supra-identity in the Asanteman and, over time, a sense of nationhood that historians once ascribed to Europe alone. [edit: The modernization and bureaucratization ascribed to Osei Kwadwo in the 1770s is also important, because it gave Asante all the key elements of a modern state, built around the same unity and identity formation.] It's a complicated thing--and its seclusion is also important, as are other symbols attached to female offices (which Jean Allman's work ably discusses)--but I'd aim you towards Ivor Wilks's Forests of Gold as one of the core texts on the interplay between Akan societies and their cultures (material, spiritual, etc) across this transformation [edit: for the era you're talking about, the 1800s]. I can only really badly simplify it here. Yes, he's an anthropologist so sometimes it gets dense for unfamiliar readership, but he also can convey ideas really deftly at other times.

I can tell you that the seclusion and control of the Golden Stool was important to Asante identity contra the British especially as their threat became more serious in the late 1800s--their demand in 1900 to produce it for their governor provoked an armed rebellion, and they buried the stool to hide it. When the stool was discovered in 1921 (I think--might be one year off) and stripped of gold and jewels by road workers, the British recognized that another revolt was imminent unless they restored the stool, returned it to the people, and allowed them to try (but not punish) the desecrators. They followed that two years later with bringing the deposed Asantehene, Prempeh I, back from exile along with the other royals. The Stool was a crucial catalyst in mobilizing and representing the nation, which is something the British only really understood after World War I. In effect, the Stool is the symbol of the nation, but not its only one.