Joyn often doesn't appear on the store if your Amazon account is set to Switzerland (geographical restrictions). However, it works very well on this Stick.
The solution is to install the "Downloader" application (available on the store), then search for and download the Joyn APK file via Google directly from this app. This is called 'sideloading' and takes 5 minutes.

Unfortunately this is not possible in Switzerland...

Yes, you can install the Joyn app on the Fire TV Stick 4K Max (16 GB). The app is available in the Amazon App Store, you need to search for "Joyn" in the Fire TV App Store and then install it. If you can't find the app, make sure your device is connected to the internet and the firmware is up to date. Once installed, you can log in to the Joyn app and stream the content.