Jeffrey Alexander reluctantly admits that some hierarchy is necessary to get the job done - but only in the uncivil spheres. The state, the economy, religious institutions, family life, and perhaps other spheres, all need functional hierarchies.
These functional hierarchies lose their begrudged legitimacy when they cross the boundary into the civil sphere. Alexander argues that the civil sphere needs to be defended from illegitimate intrusions by these hierarchies of the other spheres. Elites from the other spheres may enter the civil sphere, but only as individuals, equal with all others.
Alexander never comes out and says so, but he seems to be assuming that egalitarianism is a necessary value of the civil sphere -- in any society, not just our own. He appears to think that hierarchy is just undemocratic. He otherwise makes very sophisticated and historically rooted arguments, so it is surprising that Alexander seems to be smuggling in such a major value commitment into the very definition of the civil sphere.
I see a further problem with banning hierarchy from the civil sphere: the social movements that compete in the civil sphere need leaders. That need produces as much of a functional hierarchy as we find in any of the "uncivil" spheres.