Fascinating.
The notice Akeeba gave is clear enough, but I was at a loss where to start on this, because we neither use Cloudflare nor even have a Cloudflare account. We've got the db cluster behind a load balancer, my browser *might* be going through a proxy, and its name resolves via my hosts file (DNS is still propagating) but that's about it. I was kind of grasping at straws looking for a place to start the hunt for what's going on, and Akeeba popped up because it was the last thing on the site that changed (I updated it to 6.1.1 just before getting that notice when I tried to update Joomla to 3.8.9 (3.8.10 dropped while I was making the .9 update rounds for the servers).
It's a new server, under test and headed for production, so maybe there's something the testing hasn't shown, but I provisioned it myself and included nothing that to my knowledge invoked cloudflare. Time to put on the hard hat with head lamp and go spelunking through the system, I guess.
Thanks, anyway, Nick, for the help (sorry about the twitter duplication, I hit that first, before I even remembered the support board here). Wasn't the answer I wanted, but I'd rather an inconvenient truth than a convenient falsehood, so I'll take it.
BTW, does enabling the workarounds in AdminTools have an effect on Akeeba?