The mail log is just a log of what was sent. If there are invoices that upon edit have the email box checked, they will be sent at the first opportunity. The solution to having these all go out, is to mark all invoices in the system as sent. I think this query will do it for you, but be sure to backup your database first!
UPDATE `invoice_delivery` SET date_sent="2018-03-12 00:00:00" WHERE date_sent=NULL AND method="email";
Query will set a sent date for all invoices that have not been sent and are queued for email delivery.