Thanks Melanie but neither of those solutions fixed the issue. There is only one admin setting related to status and I think that refers to what can be seen on the advanced link of the instance settings page.
A related issue is that even when I unlock completion options and clear them all, as soon as the settings are saved, they lock again - even though there hasn't been an attempt in the meantime. I don't think this is standard behaviour either so maybe it's just that this setup is non-standard or corrupt in some way. The theme is a fairly radical one.