What causes the blackening of light bulbs
A 14 page article (https://www.jstage.jst.go.jp/article/jieij1917/6/4/6_4_398/_article) describes the blackening of incandescent light bulbs as being totally due to evaporation of the filament. Platinum had been used as far back as 1840, and Edison's first patent was for a carbon filament (1879), but by this time (1922), tungsten was the metal used in almost all filaments. 

Blackening of carbon-filament bulbs has not been a significant problem, due to the short commercial lifetime of carbon filaments. Any carbon deposition would likely be from sputtering of the filament, not evaporation. Blackening is a problem with tungsten filaments, and not so much from evaporation of metal, but from evaporation of tungsten oxides formed from adventitious water or oxygen, followed by deposition on the cool bulb surface. Inert gas is used instead of vaccuum to impede the mobility of tungsten atoms. But then, tiny amounts of oxygen or water vapor in the bulb can cause severe blackening; getters such as zirconium are used to combine with oxygen. Large bulbs used for theater lighting often contained some tungsten powder; when the bulb darkened, the bulb was shaken to abrade off the tungsten and increase the brightness. https://en.wikipedia.org/wiki/Incandescent_light_bulb#Bulb_blackening

Halogen bulbs use iodine to react with the deposited tungsten, which vaporizes as the iodide and is redeposited on the hot filament, where it decomposes. Halogen bulbs are smaller than the usual incandescent bulb and they run hotter filaments to keep the glass hotter, so that iodine can react there with the deposited tungsten and carry it back to the filament.

Your description of Hammer adding an anode to catch electrons emitted by the hot filament is correct, but I suspect that the original thinking went something like this: "We know that electrons are emitted by the hot filament. As a result, the filament must become positive; thus some metal atom at the surface might be repelled, carrying a positive charge (which would be neutralized as it passes thru the electron cloud). This metal atom might deposit on the cool bulb, producing a darkening. Perhaps we can prevent this repulsion/evaporation by making the filament negative so as to retain the positive metal ions." Apparently, this did not work, but only increased the electron emission, which could then be described as a bug or a feature.  OK, let's call it a feature and name it after Edison.