For the record, there's a fantastic third-party web site that's all about Time Machine, which led me to the proper use of the tmutil command line. I was able to reconnect the new machine to the old backups, which wouldn't have worked right out of the box.

Overall, this experience was radically less painful than what I endured when my wife's 2007 machine had a dead hard drive and we had no backup. I don't understand why the machine was unhappy with my 10.7 bootable USB stick, but at least swapping in a pre-installed drive did the trick.

Now we just have to see what goodies Apple will be announcing tomorrow (?) that will be able to properly replace this machine.