How could you knowingly break a gag order, if an automated system publicly posts all correspondence before anyone reads them?
(a) in the form of a valid court order hand-delivered to and served on your CEO at his home at 6am by two very grim federal agents who don't really like getting up this early, or
(b) if you're represented by counsel, it comes in the form of notification to your attorney by the court clerk that a court order has been issued (or perhaps notification from an AUSA if the clerk's office is slacking).
OpSec notwithstanding. If the government doesn't know your other email addresses, this obviously doesn't apply.
If everything that comes to every one of your inboxes is immediately laid bare for the world to see, they cannot circumvent the process and also cannot prove that you knowingly and intentionally violated the gag order. I don't know anyone who does this, however.