Likewise, it's not a free standing item for the same reason, and because Gygax would have used brazier (because he uses them all over the place elsewhere) if it was free standing.
Given Gygax's effusive affairs with thesaurical tomes, I would dare a speculative hazard that your deductive assay would be in jeopardous error.
Personally, I suspect it's likely that Gygax had some experience with groups scavenging for light sources in the dungeon. (Something that crops up in my own games whenever I unleash a large dungeon complex and simultaneously require meticulous tracking of light sources and their use.) Ergo, there was a fair likelihood that someone would check the cresset to see if such a light source could be scavenged (whereupon they would discover the silver baton).