On the contrary, it arrived several days before they said it would arrive. They replaced the motherboard; it now turns on properly. They also replaced the broken battery (which was not supposed to be covered by the warranty) and the power supply. All my data is still intact (and a more current backup is being created as we speak).
This does not change my mind about purchasing future HPs (it would have been nice if I hadn't NEEDED the warranty, wouldn't it?), but I am feeling less hostile to the company overall in the wake of them doing more than I expected them to do. I continue to be hugely, hugely glad I bought the extended warranty.
ETA:Once the backup update is complete, I plan to repartition. I didn't have a ton of room so I wasn't inclined to do that. But I figured since I had faced and accepted the possibility of data loss, now I am willing to take a risk in that regard. I'll delete some larger files from the hard drive, repartition, and then put them back on.