custom post types missing ADD NEW

You can’t see the menus you expect because the capabilities have not been registered. These:

$capabilities = array(
    'edit_post'           => 'edit_partner',
    'read_post'           => 'read_post',
    'delete_post'         => 'delete_partner',
    'edit_posts'          => 'edit_partners',
    'edit_others_posts'   => 'edit_others_partners',
    'publish_posts'       => 'publish_partners',
    'read_private_posts'  => 'read_private_partners',
);

All of those capabilities with “partner” in them do not exist by default. For a quick test comment that out and you should see the menus. Change “partner” to “post” and things should be editable by the same people who can edit posts (I believe that is the default behavior if you don’t pass any capability argument, but I’d have to test it or look it up to be sure.)

You can, of course, create those capabilities, if you’d rather go that route.