Why did you leave the gate area to begin with? Posted delays can and do change. They are basically estimates. Initially they thought the delay might be 22 minutes but apparently they were able to fix the problem sooner and more or less depart on-time. Personally for such a short delay I wouldn't have gone back to the Crown Room but had the delay been longer I would have notified the gate agent that I would be in the Crown Room in case the posted departure time changed so they could call the Crown Room to announce boarding.
Also, if you have access to the Crown Room why didn't you just return there to be re-booked? Once you found out the flight had departed there was no point to arguing with the gate agent and demanding a supervisor. The agents in the Crown Room are infinitely more helpful than the frontline agents. Had you just kept your cool and let the Crown Room take care of your re-booking you could have avoided the whole mess.
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