Greetings, ewkester.
I see that you're unable to download and use the Amazon application on your iPhone. Good job on the troubleshooting you've completed, and I'm happy to help.
Get started with a supervised iPhone, iPad, or iPod touch -- Make sure your iPhone isn't supervised.
How to back up your iPhone, iPad, and iPod touch -- Please be sure to have a fresh backup to keep your personal information safe. In order to back up your Health data, the backup needs to be encrypted. iCloud does this automatically, but iTunes doesn't. If you'd like to use iTunes to back up your iPhone, refer to this article: About encrypted backups in iTunes
Restore your iPhone, iPad, or iPod to factory settings -- Since you've already tried restoring your iPhone, I'd like for you to return your iPhone to factory settings. The first time through don't use the backup you just made to set up your phone. Set it up as a new iPhone. Log into iTunes & App Store when asked, then see if you're able to download the Amazon app once you're at the Home screen. If you are, restore your backup and test again.
Let me know how that goes, and take care!