Remote control for HomePod, not using AirPlay
It turns out the Music app is a fully featured remote control for HomePod. From the Now Playing screen in Music tap the AirPlay icon. Instead of choosing the HomePod as an AirPlay target swipe up and select the HomePod itself. Now tap the grey border or the selected device to close the device selection screen, close Now Playing, and select music to play on the HomePod. This isn't AirPlay, you can start something playing and quit the music app or reboot your device, and playback on the HomePod won't stop.
I have Apple Music at the moment so I see the full iCloud Music library whether I'm looking at it on my iPhone or my HomePod but presumably if you only have purchased content available to the HomePod then that is what you would be able to see if you browse the library in the last view. The Now Playing screen can be used to control volume, play, pause, skip, rewind, and if you slide it up control shuffle and repeat.
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HomePod, Space Grey