I finally did it: I have removed Google Analytics from this website and all subdomain websites.
This task has been festering on my todo list for at least a year or two. I am so happy to have this done.
Recently I read a Hacker News thread on criticising Google Analytics that prompted me to finally take care of this. The comments pointed out that many Internet users block the Google Analytics script from running, so the reported metrics both undercount and misrepresent the actual website activity. The comments also pointed out that you can get the same information by analyzing website logs, which may not provide the same nice interactive charts and figures, but can provide the same information. It just takes some time and effort to setup.
This is a static website that uses AWS CloudFront to serve the content. The logs are stored in an S3 bucket. I wrote a bash script to download the log files and create reports using AWStats (Advanced Web Statistics). Later I'll create a blog post documenting exactly how that works, since there isn't much documentation available online explaining how to do that.
Update: See AWStats and CloudFront Logs for technical details.