There are a few third-party apps (please do google/bing search) that aren’t listed on Apples’ App store.

Please proceed with caution as installing third-party apps might risk your device.

Does anybody know of an Apple TV app that runs on the TV box and can be launched just like any Apple TV app? – so I can make phone calls while my child keeps watching whatever I want to stream?

Can you elaborate on the ” unofficial alternatives”? or at least point me to the right direction?

I tried the first app on your list and it works well, but it does have an annoying problem, it doesn’t allow you to type anything on the website, so you can’t perform searches or enter your username/password in order to login to websites.
Yes, it’s quite a big problem. But, if you can navigate to the video you want to watch without using search, the video runs smoothly. I was surprised about how well videos worked.

Hi Ken, sorry to disappoint you. But, there are limited apps available in the official app store. There are a few good unofficial alternatives but it has quite a cumbersome installation process.