I found a solution to this problem that completely fixes the issue--works 100% on Amazon Fire Devices, including the new Fire Cube.
It's call Plex. I connected a Hauppage TV tuner to my Windows computer ($65), connected my HD antenna to the tuner, downloaded the Plex software to that computer, and added the Plex application to my Fire devices. Problem solved. Next, I am going to use my Raspberry Pi to host the Plex server software, connecting the USB tuner to it. The Plex software includes the DVR for my over-the-air broadcasts.
Sorry, Tablo, but you can't put your customers on hold for more than six months and still not have a solution to your problem.