I'm sure this is a false equivalency in some way, but many gov postings have wages set to the locality. My cousin was offered wages for a particular agency location, and required to move to the area.
If your cousin had the freedom to pick the specific address in a reasonably broad area, I think the worst-case obligation for that agency would be to cover the commute time from the single closest residential address to the agency's location. If your cousin picked a place farther away than that, for whatever combination of reasons mattered to them, they own the outcome of that choice (same as I'd own the decision to move to Utica).