It was nothing to do with any administrator action or settings - it was because I was on a metered internet connection. When I used the yellow "connect anyway" button in the notice at the top of the main application screen, this confusing message stopped appearing.
Unfortunately, when MS added the check for a metered network, they didn't update the UI messages accordingly. So, check your network connection and if it's metered, click the yellow "Connect anyway" button, and try again, before doing any of the other steps around editting the registry.
Many online resources for this issue in Office 2013 (example) suggest editing the registry entry
Computer\HKEY_CURRENT_USER\Software\Microsoft\Office\15.0\Common\Internet\UseOnlineContent, which has the following values:
- 0: Do not allow user to access Office 2013 resources on the Internet.
- 1: Allow user to opt in to access of Office 2013 resources on the Internet.
- 2: (Default) Allows the user to access Office 2013 resources on the Internet.
It appears that the Office 2016 "Metered connection" feature works by imitating a setting of 0 on this registry entry. Before Office 2016, this would have been a registry setting set by an administrator, and it seems like they didn't update the UI message to check it's not appearing for administrator-unrelated reasons.