Thanks for the reply c-w-s — insightful as always. But while what you say is true, it's a result of commercial choices. If BA wanted to empower their phone agents, they could, just as some other airlines do.
The other problem is the finger pointing (airport says call; telephone line says airport) and misinformation (different agents say different things). This is a training issue. Indeed, it was fairly clear last night that the infrastructure, at least at SFO, is just not well equipped to handle these events.
I'm not a first timer, and nothing about BA's IROPS process last night was mysterious to me — in fact, the whole thing was depressingly predictable! (No great problem for me in this case, but) I do think they should do better.