Using Google Home as Audio Output for STHM?

Neither ths builtin skill for Google nor Alexa sets uo the respective devices as a ‘speaker’ or TTS device by default. You need the solution Google Assistant relay: [Release] Assistant Relay V3.3 Beta

To enable access to the speaker and possibly a TTS driver like BigTalker to send speech, depending on what you want to do with it.

As a side note, Alexa doesnt require the middleman Raspberry Pi to access Echos but does absolutely require EchoSpeaks to make them show as addressable speakers im SmartThings.