I’m the author of the google skill. In theory you could host it locally (the first version actually ran on the same raspberry pi as home assistant) but it would still need to talk to Amazon servers to work so you might as well host it on AWS Lambda. This is Amazons cloud computing platform and for the purposes of using the skill it is completely free so there really isn’t any reason not to use it (you will need to give a credit card though - i’ve never been charged).
A really good set of instructions is given on this video https://www.youtube.com/watch?v=PR-LVPMU7F4&t=245s
I was hesitant with the credit card but I trust Amazon so I recently caved and set up Lambda and it has been working beautifully. Thank you for sharing this project!
I understand your position, however due the the way the skill works it really isn’t realistic to run it locally as google would very quickly block your IP as it will look like a bot (this was why I had to move to running it in AWS). By running it in AWS your requests are lost in the many thousands that must come from AWS every second.