How do I add a css class to body from a page callback function?
Well, I think your confusion comes from the D7/D6 fact that a "page" has little clue whats going on around it. As you've said the page callback function tells drupal "what goes in the $content" but not what adornments happen on the page HTML DOM around $content.

To my knowledge unless your using Panels, Context or something else to share "page state" through components of your site -- this is simply how D7 simply works.

Without using context, panels, etc. I usually end up looking for url paths or NIDs or custom content array attributes #project-add-body-classes in hook functions to fire off my logic.

For what you describe I would use hook_html_alter() and look for a page path to alter the body_classes array as needed.