Past Due Invoices in Email Log for Users

We recently turned on Past due invoice emails and noticed that they are not shown in the Users Email Log.

After talking with Support we were informed that Automated emails like this don't typically show in the email log since its an automated email. it would be useful if there was a way to have those emails appear in the email log so we know they are being sent to the users so when talking with them we can reference the email we know they received