I would presume it was because the flight was oversold that led to OLCI problems. Though it may have been WT which was most oversold, in a way that's irrelevant, since at check-in stage what matters is finding an empty seat, hence the frequent WT -> WTP -> CW cascade of upgrades. So anyone prepared to delay would have helped resolve the issue - it's an extra seat, an extra passenger who can be accommodated. Usually having status and a reserved seat will assist you getting boarding passes but if BA was unsure just how overbooked they were, and in particular if there was a lot of on-carriage (connecting) bookings which they may not have full visibility (e.g. AA bookings) then they would need a large percentage of people to pass by the check-in desk to work out what to do.