Update: This was still not loading the news earlier today.

However, I've been keeping Windows Update on pause to stop it from helpfully deciding on its own to automatically download ginormous updates in the middle of my play sessions, and tonight I decided to check for updates because it's probably been over a week since I last checked for updates. There was a .NET update, followed by a Windows update, and by the time I was finally able to start playing ESO I noticed that the launcher is showing the news announcements again.

So the issue I was having seems to be resolved, and I think perhaps it might have been caused by not keeping either .NET or Windows updated, although that's just a guess. Fortunately, the issue was not preventing me from launching and playing the game, so it was nothing more than a minor annoyance at worst, but I mention it in case it helps anyone else.