An way to do that is using SQL. You probably have access to the database, so you can run this daily. It’s not automatic, but will solve your problem
UPDATE wp_posts SET post_status="publish" WHERE post_status="draft" AND post_type="post" ORDER BY rand() LIMIT 5000
Just to explain what this line do: It will get 5000 items (LIMIT 5000) in a random order (ORDER BY rand()) and will transform the status to published (post_status=”publish”) on every record that is a post (post_type=”post”) and the status is draft (post_status=”draft”).
Hope it helps 🙂