The API:Filearchive documentation states:
Required rights none
This is incorrect, the deletedhistory right is required, see:
https://phabricator.wikimedia.org/source/mediawiki/browse/master/includes/api/ApiQueryFilearchive.php$42
This right has been required since the introduction of this API: https://phabricator.wikimedia.org/rMWe4eca2c65e3625def76d9da67950291586e6ea41