Out of curiosity, why did you choose firetv and firestick compared to the alternatives? (Apple TV, roku, chrome cast etc). I’m trying to decide
Shouting "Alexa turn the TV off" from another part of the house when it's my daughter's bedtime is worth the entry fee! Also when my wife is in the car with her Echo Auto, she can say "announce that I'm coming home" and that message will play on the TV (and convert it to text as well). Ps. we didn't know the FireTV did this when we bought it. It's quite nice finding these features as you use other Alexa devices.
I've not experienced much in the way of lagging or unresponsiveness. Note that they just released a major update.