They have one, it's just that they didn't always use them especially on ships where all the mail is placed in a bundle before being shipped to the USPS. Indeed, "MPS" mail franked with ZIP codes of now-defunct military bases are somewhat valuable to stamp collectors. I think they are now supposed to date stamp all election ballots.
Besides, for the date stamp to come into play, the ballot would have had to have been received after midnight of the 6th at the earliest for it to be rejected as arriving "too late." At the time of the initial posting, it was not yet midnight anywhere in the United States at any polling place, so how could it be "rejected as too late"?
Since it takes weeks for mail to get back to the US, and while some states go by the postmark, others go by time of receipt, some being as you say midnight before election day, others, like Nebraska, require receipt by the time the polls close, which also varies, but 7PM local time is pretty typical.
I'm not saying any of this applies in this case, but that it's possible.
I remember that from the 2000 and 2004 elections. I would think the same might apply to Army, Marine and Air Force units on terra firma, or does USPS have "detachments" in theater?