UI3 on a pi 3 or 4 should be fine if you limit the streaming resolution to 1080p or 720p. If I was going to run such a thing, I would probably increase the bit rate of whichever streaming profile I settled on (and not use a "VBR" profile unless it was on wifi), so the quality would be as high as possible for the pi hardware.
A Fire TV stick probably will have more advanced video decoding hardware and be capable of 1080p or higher. Just my assumption. I don't use Fire TV sticks.
In theory the official Android app could be more efficient than UI3, so if that suits your needs then by all means give it a try.