They did become (relatively) frequent, for me, since I think I replied that I had not been having them (as a tempting fate response to the complaint that that they were happening).
Still rare, but any at all indicates something to ideally deal with. I've already said that it's effectively impossible to diagnose exactly what, from a 500 (maybe a 503, or similar, would narrow it down; but I wouldn't know if the failover response even has any further distinction in its repertoire), at least from our side of the sever.
Could be the server self-clogging (e.g. its cach-space just needing garbage-cleaning of remnant temp files), being locally squeezed (shared resources of processing/storage, due to some separate service not properly partitioned/throttled to prevent this) or externally hammered (effectively a mild DoS, intentional or otherwise).
On the basis that someone else has already poked Tarn to draw their attention to it (ditto reporting the Announcement spam, which I didn't even realise hadn't got a Report button as I just assumed someone(s) else would have done that by the time I was made aware; glad to hear that someone has actually been proactive, though), I'm content to let those who need to worry about it do the actual worrying. I can (and just have) share a generic bit of experience about what might need to be resolved, but I'm blissfully distant from being able to actually investigate and (ultimately) fix the issues.
I know well enough the likely pressures upon whoever does have to do that (perhaps Tarn, but I imagine he's farmed out more major maintenance to someone else), and how the necessary action could range from a simple " just click/type this" to a total headscratcher, by way of "we know what to fix, but it's going to be a lot of work to do so". I wish them well, whoever they are and whatever they do whenever they do it. (And it actually makes me more relaxed, imagining all the possible work that I'm not expected to do. There's no honour amongst sysadmins, apparently!)
No actual reason to believe that the 500s have anything to do with the Announcements-spam, however. Could easily be totally separate awkwardnesses. And I'm deliberately not trying to diagnose that second issue too much, so as not to give anyone else any ideas. (I could also try my own limited pen-testing, but it'd muddy the issues at best; could even get me labeled as a "grey hat" or worse.)