Okay, taking
those into account, revised chart based on best people for particular roles based on stats:
War: Earth or Fire DB
Culture: Sapphire
Finance: Gwai
Vigilance: Opal
Public Works: Emerald or Ruby
Records: ???
Mining: Wood DB
Technically the Air and Water could also be slotted in at Finance and Vigilance, but I don't think we want a DB Intrigue advisor for any reason, and Gwai should be on the council.
That leaves, as candidates for Records, either the Air DB, whichever of Emerald, Ruby, Fire, and Earth we don't assign to other positions, and Slumbering Star (the Water DB has their lowest stat in Learning and is thus not available), plus our Sorcery students.
So we have good candidates for most stuff, I think...except Learning.