I don't know when the new update came in to my Fire TV, it could have been a while ago, although I just got the email yesterday. Good idea to keep your Fire TV active via BT, but that won't work for me because my TV is kept off most of the time and sound from TV feeds the stereo. But, so does an Echo Dot, and the stereo is never turned off so can play music from Dot audio out at anytime.
I noticed the Alexa in Fire TV said the timer was not functional at the present time, so maybe it will be there soon.
I also have noticed that the Kodi app I have installed in the (gen 1) Fire TV stick (which was a bit flakey at first) works flawlessly now and can play any appropriate file I send to it, and that could be due in part to updates of Fire TV and also, Kodi.