...And we're back.
(What? What do you mean back?)
So...what happened, as best I can tell, is that around September 9-10, another application running on my server ran into a bug that caused it to exhaust all available disk space recording far too rapid retries to the syslog.
Unfortunately, in my first attempt to figure out what had gone wrong, I was ...kind of sleepy, and got to the point of "Well, maybe I should just rebuild the instance with a larger boot volume" before I reached the point of noticing the 24GB of syslog - mainly because I thought the problem would turn out to be something else I'd already been worried about - and I wound up losing the instance reservation and having to wait until capacity was available again in order to bring it back online.
I suspect that a lot of instances gave up on sending posts to me while my server was unreachable...I wonder whether they'll just start again or if there'll be a Hassle involved.