The more I think about this, the more I cannot see a reason to change. And I think if we do start from somewhere else, there is a good chance that someone will have come in to London on the train and gone to HPC and be left there at midnight on their own, having missed the last train out of London.
HPC is near to Charing Cross for me, and equidistant from other mainline stations, so is sort of fair for everyone. And who cares about toilets? We've all gone before, as it were, and there is a stop pretty early in the ride, mostly. And shelter is OK there, although if we were too worried about the need for shelter we'd not be planning to spend all night on the bike.....