How to fix Windows 7 when System Recovery Options hangs?

I figured this out on my own. The answer is "wait at least an hour".

It turns out that System Recovery was not actually completely hanging when I got the blank blue wallpaper. It was just running extremely, extremely slowly. I let the computer sit in the blank screen state for about 30 minutes, and then a window finally popped up and I was able to choose and go through the "Startup Repair" wizard.

So, it's possible that System Recovery may run very slowly and the user may have to be very patient. This isn't documented anywhere that I could find, so presumably this isn't the case for everyone. I had to wait for durations of 30 to 60 minutes between mouse clicks. Finally, after Startup Repair ran completely (it took about four hours), it asked me to restart and then Windows 7 started up fine then.