That’s not the usual method, I wonder if you just misunderstood on first reading. You don’t add the echo as a virtual switch. Instead, you create a virtual switch which is also a virtual sensor (turning on the switch with any method will make the sensor appear open to Alexa), and then you use an Alexa routine (not a smartthings routine) which will react to the sensor opening and let you have any of your echo devices speak a custom phrase. This works very well when it works, but sometimes it glitches and doesn’t work for a few days. Here’s the community FAQ on that:
FAQ: Can I trigger an Echo Action without Speaking to It?
The zooz switch should definitely show up in your echo devices as long as you have told Alexa to discover new devices. However, it cannot be used as a trigger for an Alexa routine because at the present time no switch can. But you should be able to turn it on and off with echo once you’ve done discovery. 
Which DTH did you use?