The iPhone was alerting me that my battery might need replacement
The instructions didn't adequately explain what happens to the adhesive strip that seals the case perimeter. Some adhesive stayed with the back panel, some stayed with the front panel, and a lot of it stretched out between the two. Leaving it, as Step 2 seemed to suggest, would have prevented the phone from closing. So I spent more time than expected cleaning it up. After that, though, it was smooth sailing.
Take your time! Make a mental note of everything you do, and every part. It's a great help when putting everything back together.