I tried to connect my Xbox One S Wireless controller (latest FW) to my FireTV gen1 (latest FW). The controller was listed on the Game Controllers list on the device and after clicking on it the pairing process started the LED was blinking on both devices and then the FireTV seemed to hang a bit (it could not get input from the remote and the blinking was not continuous on the box) then the process timed out and the "Pairing was not successful" message was displayed on the screen and the Xbox One Controller was listed again on the discovered Game Controllers list.
Here I found that the Xbox One controller requires a Bluetooth 2.1 dongle to connect (which includes the SSP):
The Amazon FireTV gen1 has Bluetooth 3.0 dongle integrated, so in theory it should work with the controller.
This is the log during the pairing process: