Replying to my own question. I finally got it working. It turns out client-download.steampowered.com uses round robin DNS and has multiple IP addresses. Murphy’s Law being what it was, my system of course decided to pick and cache the mirror that’s not online. So, I put one of the working IPs in /etc/hosts, did a quick ‘systemctl restart nscd’ and retried. Bingo!
Putting this here in case anyone else has this problem. It was very frustrating.
At the shell prompt, do ‘host client-download.steampowered.com’ and ping each of the IPs it returns until you find one that works. Then add, for example, the following to /etc/hosts and do a ‘sudo systemctl restart nscd’.
/etc/hosts: Add:
162.254.192.11 client-download.steampowered.com