F3G - Aside from the totally selfish attitude in your remark about "who cares if economy pax are pis*ed off if one high payer arrives a few minutes late", those few minutes may well lead to the flight picking up a delay, which may lead to huge inconvieniance and cost for the airline (especially at a busy airport like LHR) - it could push the crew over hours, mean a delay for the next sector - anything really.
I am not a huge fan of BA per se, but I do see that they are trying to turn around their performance.
I just cant see the problem - if you arrive late you dont board, solution? Dont arrive late.