Updated answer:
Let’s modify the corresponding dashboard query so it returns no draft posts.
Example:
If I recall correctly the default Hello World post has ID=1 and the default Sample Page has ID=2.
So if we set ID=2, then we can remove the draft list, since it is fetching the post post type.
Since WordPress 4.4 (see #8243) the dashboard_recent_drafts_query_args filter is available, that makes it easy to target only the relevant query:
/**
* Remove recent post drafts under the Quick Draft on the dashboard
*
* @link http://wordpress.stackexchange.com/a/201910/26350
*/
add_filter( 'dashboard_recent_drafts_query_args', function( $args )
{
// Adjust parameters to your needs to return empty results
$args['post_type'] = 'post';
$args['p'] = 2;
return $args;
} );
Hopefully you can adjust this to your needs, with other parameters if needed.
Another way would be to use $args['post__in'] = [0]; to generate the wp_posts.ID IN (0) WHERE part. I think I’ve seen this approach in some answers here on this site, but don’t recall where. We get the same result if we use null, '' or even false in place of 0 because it’s taken through absint() via array_map().
This is the result:
