Why doesn’t HomePod mini work with Apple TV?

It does work

Use Apple TV to play audio throughout your home - Apple Support

• Press and hold the TV  button on the Siri Remote to open Control Center, then select the Audio Controls button .
• If you see the Audio Controls button  in an app, select it.

A list of available audio devices appears, organized by room (as assigned in the Home app on your iOS, iPadOS, or macOS device).

The privacy setting is showing you which third party accessories are using homekit 

There is  no Homekit app. HomeKit runs all the smart devices, turning on lights etc. 

Home is an App. Can another be substituted? No. It’s an Apple App using your Apple ID and you wouldn’t want to trust a third party App with all your details . However smart devices can also have their own apps you can install that run their lights etc, but aren’t using your Apple ID. They are good to use additional features the Home App doesn’t include 

Once you set up, change or do anything in the home app one one device it will sync across all your devices signed into your Apple ID that has the Home App. It’s not the one is better than the other. It’s so you can go to another device to control things. Apple TV does not even have the Home App, it’s a Hub.

 Basically it’s your Apple ID being used to sync everything. That is how it “knows”. Sign out of your Apple ID on Apple TV and  see how much it knows then. You will see Apple TV prompting you to set up,HomeKit 

It’s hard to explain and the only way you learn is after using it for awhile. I probably confused you even more. I can’t explain it very well in layman’s terms. I think I confused myself writing this out. It could be improved on but once posted the edit window will vanish in minutes 

I’ll leave you with some links: 

Set up and use the Home App

https://support.apple.com/en-ca/HT204893

Create home automations with the Home app:

https://support.apple.com/en-ca/HT208940

HomeKit vs Home

it is a fundamental question, Privacy controls under settings on my iPad has an entry for Homekit But NOT Home. Under settings main list Home is listed but NOT Homekit. Accessories et al are clients for Homekit and Home App can control on behalf of Homekit, furthermore, AppleTV and iPod know Homekit intrinsically and do not need any interference by the user because they know it it all and everything, inclusive, is optimized in 5he best interest of the Homekit owner’s Home.

Are you confused? I bet you are not, but I am lost:

Is there an App clearly identified as Homekit?

Where are the settings for Homekit?

if the Home App is a client of Homekit, can it be substituted by another app?

I have been attempting to understand the Apple Brand integration of everything Apple, and I have failed consistently. In this case, the Homekit vs. Home, illustrates my total loss in understanding everything Apple.

You’re welcome