So, I put together a Bash script, which I've published at, which can do the Chromecast initial setup and connect it to WiFi without Google Home or an Android or iOS device. There was also some useful information at about how you need to encrypt the WiFi password with a public key from the device before submitting it.
However, I did find some documentation of the Chromecast's API at, and I noted that if you're doing initial setup, none of it needs any authentication.
I had no luck with answer, because I could only find Linux builds of Chromium, not Chrome, and Chromium seems to never have implemented chrome://cast.