You can just run any old DNS server with IPs that you can only access if you also happen to be on a particular ZeroTier network. Route 53 works.
The tricky part of ZT-only DNS would probably be making it work for more than one network at once. So you couldn't send all DNS to one place that serves *.myzt.example.com and recursively resolves anything else, because it wouldn't know about myzt-2.example.com.
Edit: a Route 53 map of your network may be easier to achieve if you use the terraform provider, which I have let languish for a while and should probably give to someone else to maintain (ZT people -- want to take it on? You've got Go expertise now!): https://github.com/cormacrelf/terraform-provider-zerotier
I can confirm I have an android TV with sideloaded zerotier (would be great with official Android TV support, BTW! Ed: Android TV has a separate app store - apps need to declare themselves compatible), a few Linux boxes - and mDNS/bonjour works fine.
The tricky part of ZT-only DNS would probably be making it work for more than one network at once. So you couldn't send all DNS to one place that serves *.myzt.example.com and recursively resolves anything else, because it wouldn't know about myzt-2.example.com.
Edit: a Route 53 map of your network may be easier to achieve if you use the terraform provider, which I have let languish for a while and should probably give to someone else to maintain (ZT people -- want to take it on? You've got Go expertise now!): https://github.com/cormacrelf/terraform-provider-zerotier