I've made no progress on this project since the last post almost a year ago. At that time, I found a Samsung Stratosphere (an early Galaxy S model) in a drawer and started using that. (I had originally found it in an electronics recycling bin at school.) It's running Android 2.3.6, basically what my first phone ran, but I've managed to get used to not being able to run modern apps. I still have no cell service, so I'm effectively using it as a tiny tablet with a built-in keyboard that I don't use. Also, I haven't taken my own advice at the very end of the last post.
Anyway, I realized what might have been keeping the tablet from booting. Many devices take considerably more current when booting than they do to run after having booted (…I think). The battery I installed is much smaller in physical dimensions than the old one. Therefore, even though its capacity is greater than that of the degraded old one, it might not be able to source as much current as the old one could (before it was completely degraded, anyway), because that's a function of the physical surface area of the electrodes inside (AIUI). So, when I get around to working on the tablet again, I'll try connecting it to a bench power supply. If it can boot that way, then I'll just remove the interim battery and go straight to the 18650s—those should be able to give it enough current.