The used iPhone 5 that I received from a friend was stuck in a bootloop and nothing brought it out. I tested with a known working battery and the phone booted right up. I knew the battery was to blame then.
The repair was easy enough, I've done it many times by now. The replacement battery worked great for 2-3 days until it refused to charge and just kept dropping in percentage even while plugged in. I contacted support and after some discussion, they sent me another battery completely free of charge without even asking for the defective one back, even after I offered and expected to send it back. The second battery works great so far at just over a week in.
UPDATE: This review was written on July 4. As of September 10, 2017, my phone kept shutting off randomly for the past 2-3 days and as of this morning, it won't start and won't respond to any buttons, even the 10 second home + power button press for hard reset. Plugging it into power results in an Apple logo for a few seconds, then a black screen and repeat ad infinitum. So it's back to the original problem that lead me to replace the battery in the first place. I will contact support after this update.
If the battery rattles slightly from side to side inside the phone, add a tiny piece of the plastic baggie the battery came in to cushion it and prevent it from moving horizontally within the phone.