I think "Chromecast with Google TV" excludes older Chromecast devices. Not that I really blame Apple here, I've played around with Chromecasts a few times and they're wildly underpowered even if they are ubiquitous.
The appleTV remote couldn't change the volume on my tv for whatever reason. Since I bought a dumb monitor style TV the only option was to walk up to it and use the buttons. Switched to chromecast and the volume controls worked. Hard for me to imagine going back to apple, even if the chromecast is a tad slower.
The Apple TV remote can be configured to control volume via IR or through HDMI CEC. When plugged into a new display, the Apple TV tries to auto configure this based on what devices it detects. It's possible it messed up with IR and you just needed to manually set it to use CEC.
