FYI You are now able to use an Apple Music Subscription as your music “service” inside the Google Hone App.
If you have Google Assistant on a smart speaker you should be able to command it to play music using your Apple Music Subscription.
Once you have named and added the chromecast device in your Google Home app and set the music provider, you should also theoretically be able to use the iOS Google Assistant app to tell it what to play but I currently can only get it to work atm asking my Home Mini as an interpreter between my Apple Music Subscription and my Android TV & Chromecasts.
It’s dumb that Apple refuses to add cast integration, especially with how desperate they are to sell AppleTV subscriptions today to compete with the bigger players in that field. Surely the more platforms you allow it on the more you reap from your investments in creating the media. An AppleTV box is a one off profit but a loyal subscriber on any platform is a steady revenue.
-But at least Google have opened the door on their side allowing it as a music service on Google Home.