Dialog "Amazon AppStore Required" - apps prevented from starting
The solution is as stated, though it was not easy: re-install Amazon AppStore. Amazon Appstore is not distributed from Google Play any longer (though it was when I first installed it). It must be installed from amazon.com (as a 3rd party app - Settings &gt; Security &gt; Unknown Sources must be enabled), and updated to the latest version - Amazon Underground for the dialog to stop appearing when starting "Free App of The Day" apps. This left the device with an unwanted app with necessary space consumed unnecessarily.

After the installation, I was able to test the correct opening of certain apps I got through the FAoTD program. I tested starting them both ways: from the "Your Apps" list from within Amazon Underground, and outside that environment, directly from the launcher: 


2Do (which had a lot of checklists for Books and Movies I would have lost),
DoggCatcher,
mSecure,
Monument Valley, and, I presume, the re-enabled startup of
Nearly 2 dozen other apps and games.