Only two or three files are served from Cloudfront, and I'm not aware of any analytics software that reads those logs. If there is one, please let me know, I'm interested.
Sorry I don't know much about this sort of thing. It's just something that seemed conceptually possible? It's no mystery why Google would want all the data GA creates, but I've always wondered why e.g. Piwik isn't more popular.
Piwik isn't popular with me just because I don't want to run it myself (and keep it upgraded, etc), and the hosted version is as bad at tracking as GA is.
We send all of our logs to CloudWatch Logs and then set up metrics for 200, 300, 400, and 500 http responses. It will graph all of them for you right in the console and you can add alerting if you need it.
