"Bypass content access control" really means let users in a role with that permission view unpublished content (and edit and delete any content, unfortunately). Administer content does not grant this power to see unpublished content nor to edit and delete any content. In Drupal 7 this permission seems to do little more than what the separate "Access the content overview" page does.

This bypass access control permission is related to content only (nodes, hence why it is in the Node module section of the permissions page), not to any kind of access to administration.

The tricky part is that without this permission a person may have access to edit a content type, and to "View own unpublished content", but if they edit a post originally authored by someone else they get access denied when they save it. Hence the need to give the over-powerful "Bypass content access control".

[@TODO confirm statements above, and probably file a bug report or feature request.]