Sometimes, airlines insist on the Crew doing their own loadsheets. This happens a lot on ad-hoc or wet-lease aircraft. If the flight Deck were doing this on the Mint Air/TCX flight, I would expect the crew were probably unaware of the balance issue till pretty close to pushback time, so sorting out the issue prior to boarding was probably impossible.
I have seen occasions (though usually on smaller aircraft) where passengers are moved just to make the paperwork correct. The aircraft is still safe, but flying pretty close to its trim limits, so to be on the safe side, a small amount of weight needs to be shifted. I travel on the Jetstream 41 regular, and this is common. I have even seen sandbags placed in the hold when loads have been light!
At last minute, its easier to get passengers to move to the rear than to get the equivalent weight removed from one hold and moved to another. Getting beltloaders etc takes time and adds to the delay. Remember that sometimes it can be as little as 250kgs that needs shifting. On some flights that could be 3 men (using standard EZY weights) or 20ish bags. Its not difficult to see which is easier to move.
The mistake was mentioning the faulty door. The captain should have had a quiet word with the crew, and the crew should have had a quiet word with the passengers. "Excuse me sir, would you mind if we moved you and your family to row 21, theres been a problem with the checkin system", is a lot better than "We've had to put all your bags up front, because the hold door is faulty, so to balance the plane, you need to move back". The latter case would un-nerve me, and I am a very experienced traveller.
RTG!