http://www.techradar.com/news/firefox-has-come-to-amazon-fire-tv-devices-bringing-youtube-access-with-it

Mozilla has announced that its Firefox web browser is now available on all Fire TV devices in all countries, joining Amazon’s own Silk browser. Navigating web browsers on big-screen TVs isn't the most user-friendly experience, but for Amazon Fire TV users it could soon be the only way to access YouTube. 

Though there’s no explicit 'hey, this is a convenient workaround' section in Mozilla's announcement of the news, there is a section of the blog post which states that users can "go to YouTube and other sites directly from the Firefox for Fire TV home screen" and another which promises access to videos from "YouTube and other popular sites".