Syncthing is showing as Android TV compatible on the Play Store, meaning I can download and install it on my Shield TV.
An app presenting itself as compatible suggests you can use it with either a remote or a game controller. However, Syncthing isn’t useable at all. I can’t access any of the menu items in the navigation drawer, meaning I can’t access the webui, restart the service or even access the app settings.
Some further testing needs to be done here, and maybe a look at the UI for Android TV too. In general you want it to be very clear which button is currently selected, the syncthing colour scheme doesn’t really allow for that. While that’s great for a screen that you tap or click on, it gets difficult when you need to navigate the menu yourself.
I’m happy to run test apks on my Shield TV if you don’t have an Android TV.