Tolkien’s anarchism was a decentralized voluntary association where one would swear fealty to a king and issues became common and local among people. This system was reminiscent of the Shire in his books. He supported monarchy because it acted as an involuntary position based on the catholic principle nolo episcopari. He believed this would avoid the cutthroat nature of party politics and leave the king as a figurehead that has respect and authority through voluntary exchange and respect for the monarchs position in the tradition of the country.
That’s not exactly what it means: http://wikibin.org/articles/anarcho-monarchism.html
Given that definition we wouldn’t exactly call them powerless.
Edit: This Reddit thread shines light on both ‘Anarcho-Monarchism’ and J.R.R Tolkien’s beliefs: https://reddit.com/r/monarchism/comments/pvkkjy/what_is_anarchomonarchism/
from that thread , see emperor norton or notes on Tolkienism