I got it.
Mine is the HTC One Developer Edition. Android 4.3.
I discovered, if I turn off the Developer settings, then reboot the phone, the Developer settings are automatically turned back on.
With that background, here are the steps I performed, which got the Kindle app running on my phone:
1. On the Amazon website, I made sure my phone was deregistered.
2. I removed any Kindle app from the phone.
3. I turned the Developer settings off.
4. I restarted the phone.
5. Turned the Developer settings off, again.
6. Reinstalled the Kindle app, from the Google Play Store.
7. The app ran correctly and I was able to register the phone, at which point my library synced as normal. I confirmed I can open and navigate books. When I close them, the app correctly syncs my farthest location.
8. I closed the app and found I could turn the Developer settings back on and the Kindle app runs fine.