Similar to Apple relying on Airplay for sending their audio around, YouTube Music is leaning on Chromecast for this. Similar to the plugin Dan mentions above, there may be a Chromecast bridge plugin but will certainly be a 'YMMV' situation. Quality-wise, I suspect it would not be greatly different from Airplay.
You don't need to worry about Airplay or Chromecast being a negative impact on quality, as YouTube Music maxes out at 256kbps anyway which is well within Airplay/Cast bandwidth.