Actually they do work with Dish Anywhere apps: on all THREE of my various geneartions and models of Fire Tablets, including one from 2013. Fire Tablets support both Dish Anywhere and Slingbox. In fact both apps work excellent on the tablets. Both apps are at the Amazon App store and are labled as being for Amazon Fire. Also, the Amazon Fire TV (Amazon's Roku-like box) can run the Sling Player app connected to any TV. It works very well.
BTW, Fire 10 is a very good inexpesive tablet if one wants a big screen and good specs. While the no longer manufactered (AFAIK) HDX models has supreme pixel density and color accuracy that displays high quality video in all it gorgeous nuances (at least as good as the more expensive tablets like Samsung, etc.), the Fire 10's lower pixel density and slightly lower color accuracy/reproduction still makes for a very good big screen value that's easy to read and works with both Sling(box) Player app and Dish Anywhere. It also makes a great Senior Citizen tablet (easy to read) with voice dictation for emails and searches, etc. The Fire tablets offer the best value with very good specs. and performance. However, Fire Tablet is a forked Android so not all android apps will load on it, but the most important ones do. However, depending on one's usage, it is wise to have at least ONE full Android tablet for some less popular but useful apps one might need. I've got to say that I just don't use my full Android tablet that much anymore. I'm 98% on my Fire's.
FWIW, if you don't like Amazon's Silk for web browsing (just as invasive as Chrome), Firefox can be loaded as apk file. While Firefox provides the most privacy, I use it along with extensions for enhanced privacy, ad blocking, reduced tracking, blocking 3rd party sites, and cookie killing, etc.

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