Amazon has officially removed this Fire Stick capability forever

Vega OS will run on all future Fire TV Sticks, which don't support sideloading Android apps.
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Amazon goes all-in on Vega OS for Fire TV devices (Android based Fire OS media streamer’s days are numbered)

Amazon has released two new Fire TV Sticks in the past two years… and both have shipped with a Linux-based operating system ...
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Amazon’s new Fire TV Stick HD is faster, thinner, runs a more locked down operating system

The new Amazon Fire TV Stick HD is a $35 media streamer that plugs directly into the HDMI port of your TV, allowing you to stream video from Amazon Prime Video, Netflix, and a bunch of other sources.