But: why is a Captain (or the rest of his crew) of a transport category aircraft unable to calculate the maximum allowable flight-duty and ensueing rest-time by himself? 
Even if the flight has been crossing several time zones.
The crew was well able to calculate their duty time. The clown at the other end of the line was just playing the fool when he said the crew wasn't out of time. As the text states, if you've only got 3 minutes of duty time left, for all intentions and purposes, your duty period has expired.