At airports like London Heathrow the police who have airside passes are there partly for protection against terrorist attack and so are armed. So it's not that the police are chosen to attend aircraft for immigration or drugs policing are chosen because they are armed, but because they have airside passes and are in the airport.
That is also why when a passenger starts arguing a bit too much with the airline staff and won't calm down, armed police arrive. They don't need the arms for this problem, but they happen to be armed for their other duties in the airport.