Bungie has now acknowledged the problem in its FAQ. It’s little more than “we are working to figure it out” but, when you’re otherwise hopeless, this is good news.
However, Corey at work started getting disc read errors too, and his were on a freshly-transferred Elite system, right outta the box. Paul found him a simple workaround that seems to be holding for him, and it’s working so far for me: Don’t continue your game using the “Resume Solo Campaign” option at the main menu. Instead, load up the game from the Campaign menu, as if you were going into a co-op mission. Just play from the beginning of the chapter you’re stuck at (in my case, “The Ark”). The load was very slow, but it load worked. I am watching the opening cinema to that chapter now and I’m able to play a little bit, but if I die, I get a DRE when it tries to reload. Also, I get a DRE next time it tries to load new content, after my first battle area.
Yes, I feel stupid for not trying this option before, but even though I can only get about five minutes further into the game, it gives me greater hope that this problem can be completely eradicated with a title update/patch file. This is a fine manual workaround in the meantime. If you have this problem and Paul’s advice works for you too — or doesn’t! — comment and let me know.