The writeapi right is not manageable through Special:ManageWiki, it seems to be manageable with restricted permissions only (Stewards). By default, the writeapi right is assigned to everyone. But someone might like to restrict it, for example, to bots and sysops.
All users, including anonymous users, have that right. Limitation of writeapi permission could prevent vandalism through the API, e.g. on large wikis. The defaults should remain but it could be changed on a particular wiki.
The writeapi permission is also a required component for certain features and extensions, some of which may not be immediately obvious. This includes VisualEditor and DiscussionTools, though other extensions such as Comments may also not work without that permission. That being said, it's hard to determine the entirety of what may be affected given that some of these extensions don't list the requirement.
Limitation of the writeapi permission could prevent vandalism through the API—extensions such as VisualEditor and DiscussionTools also may be target. If there is needed to mitigate the results of that, the writeapi permission can be temporarily removed from some user groups, but the edition of pages via the "edit" tab to be still available to everyone.
It was intentionally restricted from ManageWiki because of a ton of reports with things not functioning at all if it is removed. Even if it was removed from just * adding it to another group did not actually fix the issues, it had to be in *. For this reason, I do not recommend we reverse that, and allow it in ManageWiki again.