According to the BT chap that came round, that is the cause. Something to do with separate circuits.
That would explain why your father's BT Box wouldn't work properly on the extender, but I don't see how it could damage the BT Box or how replacing it would be helpful.
My guess would be that the connection between the extenders was poor and after a while they would 'unpair' themselves. Perhaps every time the BT Box was replaced the extenders were reset giving the engineer enough time to get out of the house?
Of course you may have tried resetting the extenders and rebooting the BT Box and found that didn't help - so the problem lies somewhere else. I still don't believe the extenders could permanently damage the equipment they connect to though.