Conflict between Capabilities and Menu Visibility with Custom Post Types

You probably cannot see the new CPT items in your menu, because you have indicated that they require specific caps, yet you have not assigned those same caps to any role – including your own.

Add the following to your code:

    function my_cpt_add_caps() {
        foreach ( array( 'administrator' ) as $role_name ) {
            $role = get_role( $role_name );
            $role->add_cap( 'edit_assignments' );
    // add selected other caps here
        }
    }

register_activation_hook( __FILE__, 'my_cpt_add_caps' );

On plugin activation we assign your new caps to the admin role. Now, your CPT menu items should be available in the menu. Also, only now will plugins such as Members see these new caps, as a cap must be assigned to some role before it will be acknowledged.

As you are using capability_type in your CPT registration, you can drop the capabilities array, as WP will work this out itself.