Responding here now that things appear to have settled down -- it looks like there was a firewall blocking my server's access to the outside world, and since I'd never experienced that before, I didn't recognize the issue. My only working theory was that the Ubuntu/Debian server I was trying to connect with was mis-configured, except a) that mistake's very unlikely, and b) it would have been spotted very quickly, and fixed. That left a configuration problem at my end, and the server was set up at home (where this problem didn't occur), and then moved to a client site -- thus, the issue was with the client site.

Hindsight's 20/20. Thanks for reading!