Hosting my websites on Linux environments, I never stopped using server-side generated statistics (based on Apache's access logs) with tools like AWStats. Used together, I feel that client and server-side stats give a much better image of what really happens on my websites.
But in the end, I feel awstats isn't enough, and I'll take a look at other solutions that have been pointed out in these comments. Thanks!
You do give up the ability to get live stats, but you get better performance and the ability to track more visitors (read: people like me who have blocked GA for years...)
Added bonus: Referrer spam is automatically blocked by default.
https://piwik.org/log-analytics/
EDIT: Almost forgot GoAccess, if you are okay with a terminal app and want live stats (can also be scripted to generate HTML reports) - http://goaccess.io/
Btw, tail -f /log.log is always fun for live data...
Also, please consider adding your Google Analytics use to your privacy policy. They require it, even though most people ignore this requirement and there is no enforcement.
Side note, I just tried @ replying to your companies post on Twitter and got a @your account may not be allowed to perform this action' http://i.imgur.com/9uoEn9B.png Strange, never saw that before...
That's an important point; I'm seeing referral spam in my GA reports which I never noticed before 2015. I manually discount it when compiling reports, and my understanding is that these spammers hit GA UAs at random without even loading your website.
The same way you can gather all available stats on server side and push these yourself. You can override IP too. Obviously not everything will be available, like screen resolution, but you still can capture a lot of stuff.
I've personally never even tried using server-side analytics because I assumed there would be so much noise it wouldn't be worth the trouble.