"the alternative spelling of 'archeologist' which has its own entry in Chambers" suggests "archeology" has a separate entry from "archaeology".
I've not looked in my physical copy of Chambers but searching it online for "archeologist", it's not a separate entry.
The word is included in the heading of, "archaeology or (US) archeology".
I'm not convinced that "Egyptologist", with the connexion to a form of housing, pyramids, "just did not fit the clue."
(I have no interest in the outcome by the way)