For 20 bucks you can get an hdmi audio extractor[1]. So for a little bit extra money you can simply connect the regular chromecast to your amplifier.
[1] https://www.amazon.com/slp/hdmi-audio-extractor/xc32yz93wutk...
Probably the only solution but it isn't pretty. Can't find anything like that locally so it will be easily the cost of the chromecast just for that.
And I can't find anything about whether it will resample the audio (dealbreaker), or how it will negotiate that.
I'm sure there will be a couple of unforseen issues as well. My chromecast occasionally disconnects itself, which means that you would have to rewire the hdmi cable to the TV just to have a clue of what is going on (can my parents or house gets figure that out when I'm not around?).
And yet another device and power brick.
Not that the chromecasts are a wonder of ingenuity or anything, it's just that they are stupid enough and big enough to get software support. Nothing else is comparable.