I wonder if perhaps the closed captions on the video that doesn't show it, has been encoded in a format not supported by your Fire TV, but is by the program you use in your PC.
These quotes are taken for advice to developers wanting to work with Fire TV. (see ''https://developer.amazon.com/public/solu... )
Fire TV has limited platform-level closed captioning support at this time.
Amazon Fire TV includes limited platform fonts.
Maybe you should check if you have the latest software installed. It should be automatically updated, but here is a link to show you how. Scroll to the bottom of the page. https://www.amazon.com/gp/help/customer/...
Here is a link if 'updating' causes a problem.
Perhaps the only way to prove it, is if you know someone who has the same model Fire TV with the same software version installed and see if it works on their Fire TV. If it fails when viewing the same video, it is a Fire TV, certain video CC encoding, compatibility problem, if it works it is your Fire TV.
Hopefully this may be of some help.