Knowing that the Original post is asking for something the echo itself will not do, I have found (since reading this the other day) that I can do what the OP wants, with the Harmony hub, by voice. The Harmony hub product can be purchased with or without a separate remote. I have the stand alone hub. With that set-up with my Echo system, I can ask Alexa to turn on/off any item and also set up routines such as "watch TV" or Watch FIRETV... which is pretty much what we're talking about.
I believe the Fire TV would be something to explore but I haven't any experience with it to corroborate it as a good device to achieve the desired results. Could be...
I originally set my Hub up to control pretty much everything in my entertainment system, but found the kids (grand kids) cannot keep their mitts off the buttons and screw up the Echo and hub's Status memory, which is the biggest drawback of the Harmony hub and all Video that is not "smart" already somehow. I removed all of the Game consoles from it and no longer use them. I also ditched the Fire TV stick due to it's poor performance, and eft the Chromecast and the AppleTV hooked up and have far better response and performance of my systems *I* use.
The Hub assumes what you are asking it to do is possible and sends out IR signal to turn on/off equipment just like a normal remote control. If someone manually turns an item on or off using the buttons on the equipment (or even the supplied remote in most cases) the Hub thinks it's still in the status it last left it in... One day more of our average home equipment will offer Smart Status outputs and these things will be taken care of, but until then...
Anyway, that's just another way of using your echo to operate some Video equipment as well as the audio it is connected to. We don't use Alexa for Volume control or any other functionality that it is capable of... just the on/off and select which we choose to watch. I know there's a ton more functionality in it, but we're pretty easy with our choices.