As a further update, I’ve copied all of the City of / Village of / Town of names into the P1448 (official name) property in wikidata. And then, I reversed that data back down to the official_name property in OSM along with P1813 to short_name where it happened to already exist. This grabbed a few extra data elements where those properties already existed, including a few cases where a member way of a boundary relation had a wikidata tag and corresponding properties.
This change is in this changeset for reference.