just to say that I absolutely love your thread title - hilarious !
You seem to have had a real fast track experience at LHR T5 - my congratulations, since on 90% of my travels I duck under the cordon (often to be yelled at by staff) to join a non-fast track line which is usually much shorter.
Until recently I would have agreed with you on the non-caring about missing connections. However last month I was very positively impressed, receiving a call in the afternoon that my YUL-LHR was significantly delayed, and proactively offering me a rebooking on an AC nonstop to my final destination. I had the eticket in my email inbox 5 minutes later. An awesome experience. Since the BA flight was really only about 1 1/2 hours late (unfortunately not unusual for the YUL flight) I was wondering whether perhaps they were overbooked as well which could have triggered their generosity....but sometimes it's better not to know to not spoil the excitement.