It does indeed look like an SSL problem, but on the server side.
It seems like I’ve gotten the error rate down to zero by hacking the HTTP library my bot uses and forcing it to send only one request per connection, instead of trying to re-use connections. Persistent connections were working perfectly well for 2.5 months (since the bot first came online), and then 10:45pm EST last Thursday happened. I don’t know what changed, but it wasn’t on my end.
That said, persistent HTTPS connections should work. To not allow them has been considered bad practice for at least the last 5 years.