Workbench is a great way to do this if you are OK with installing another module.
Workbench allows you to add moderation states. By default it gives you 'draft', 'published', and 'needs review'. (Drupal 8 also has "archived"). There is also a set of transitions that go with these states (ie, draft -> needs review)
For the three roles you described above, I would give the Content Editors the states to move from published -> new draft, draft -> needs review, needs review-> draft. Content Publishers would have access to all transitions, and super admins could override workflow entirely. This would produce the result you are looking for (and is what we commonly use with clients who don't need something more complicated).
D7 workbench can be a little confusing (very powerful means there's lots to configure!), but here's a good quickstart guide:https://www.youtube.com/watch?v=8nqWLrxH_4M
EDIT: As the OP points out, Drafty is a dependency for Workbench Moderation, which is needed for the above described solution. Drafty is no longer supported under Drupal' security advisory policy.