I have Spotify Premium and use the Connect function already on the Playstation 3. There also works wonderfully in super quality and absolutely trouble-free. Thought that it will work equally on the FireTV stick. FireTV is indeed listed in the Spotify app on the smartphone but then the connection fails with an error message. I very much hope that will be touched up here and a similarly good app like on Playstation appears. There you can also choose the music the way without smartphone, which is a further advantage over the FireTV version.
Supplement: There is now an update that has to be triggered manually, however. Thus Spotify Connect works so far quite well and I have corrected the rating upwards.