If running even a little behind schedule, I think BA should get the aircraft loaded as fast as possible, and that means allowing anyone who gets to the desk quickly onto the plane, instead of sending them back.
This will wind up a lot of people who feel they aren't getting what they are entitled to (and I would be one of those), but I will happily stand in a queue (or even a minor scrum) for a few minutes if it means arriving at my destination 5 minutes earlier.
However, when everything is on schedule and there is plenty of time, BA should enforce their rules, if only to avoid annoying people who are aware of what the priority rules are.