I think renfrew has hit it on the head. Too many "mistakes" are being made these days and too many pax are just plain dishonest about it. Ascribing fault to the web interface isn't really an answer because you are given the dates and times of the flights on so many successive screens during the booking process that you really shouldn't manage to overlook the fact that you're booking the wrong flights.
However, if the agent genuinely accepted that it had been a genuine mistake, and if there was genuinely an open seat on the aircraft and that it was not overbooked at that stage, then it is a bit inflexible for an exception not to be made. But inflexibility on these things is part of the delicate balance between revenue protection and customer service - and it's obvious that with BA's lower fares must come stronger revenue protection.
It's at this sort of time that frequent flyer status can work wonders (although never guaranteed). An entry level card probably won't get you anywhere - but I'd be surprised if this had happened to a BA Gold.