Just looking at what went wrong - BA staff giving wrong information in the lounge - this wasn't malice or stupidity or even lack of training. The answer they will have given "wait until a gate appears and relax until then" is the correct one almost always.

The boards are not unreliable in any general sense so far as gate display goes, and hence the suggestions to turn the screens off or put disclaimers on them or rewire to use the Heathrow feed seem rather disproportionate. May not even be an IT issue, it's arguably more likely someone forgot to update the gate. The staff were likely as wrong footed as the seven unfortunate travellers by an unexpected and unusual event. It's easy to be supercilious when these things happen, but let whomever has never made a mistake cast the first first brickbat.

And the bottom line is that if you only ever rely on the lounge displays and nothing else, your chances of hitting the same problem are negligible. I think these things hit the "what if this happened to me?" part of travel psychology, and people over-egg the issues.

In all fairness, the OP seems to have accepted the advice that it's not really worth pursuing further - this would be in my experience akin to trying to win a fist fight with a blancmange. BA have done a decent job on service recovery, they rebooked and paid compensation, and implicit in that is an acceptance of some sort of blame. But you will never get a business of this size to admit more liability than they strictly need to in a written communication, even if individuals are sympathetic as seems to have been the case at various points. Just not going to happen. Take the money and move on.