To correctly show the Excerpt meta box for Custom Post types you should use the supports property register_post_type. Supported values are as follows, and I suggest referring to the register_post_type Function Reference for more details –

• ‘title’
• ‘editor’ (content)
• ‘author’
• ‘thumbnail’ (featured image, current theme must also support post-thumbnails)
• ‘excerpt’
• ‘trackbacks’
• ‘custom-fields’
• ‘comments’ (also will see comment count balloon on edit screen)
• ‘revisions’ (will store revisions)
• ‘page-attributes’ (menu order, hierarchical must be true to show Parent option)
• ‘post-formats’ add post formats, see Post Formats

The purpose of add_meta_box is primarily to "allow plugin developers to add meta boxes to the administrative interface". So really you should only use it to add your own custom meta boxes, and there is no need to use it to add built in meta boxes.

As a side note, the docs for add_meta_box also note the following –

This function should be called from the ‘add_meta_boxes’ action.

Update

Also, please check that the other users have selected to show the Excerpt box, as I believe it is hidden by default. To do this click Screen Options and then ensure that Excerpt is ticked.