You assumed that they mass emailed the notice and are incorrect.
As the article states, the notice was sent by "cable" which is the old telegram system and still the only official means of communication between the Department and US Missions overseas.
The cable system is on a completely separate classified network.
As the unfortunate recipient of the mail storm emails I will say that many people included information in their replies that referenced the cable (and subsequent Department Notice) telling people to stop hitting reply to all so you are not entirely incorrect. It is just that the Department was smart enough not to send out a blanket email to everybody.
The other thing that seemed compound this problem was that the To: line didn't have any names or mailing list groups listed. People (idiots) didn't realize that they emailing almost everyone in the Department.
I would also point out that the email servers slowed but I never experienced any lost email or service interruption. Some emails were delayed by as much as two hours.