Well, actually, it does. Because Android to be useable requires Google account.
No.
I very deliberately did NOT set up a Google account on my Android Fairphone, and it does the basic things just fine, like, um, phone calls and even alarms. It even takes OK pictures.
I have EU citizens' contact details in my phone and I think that, given NSA revelations, I would be breaking the law to knowingly share/sync those details with/via a US entity such as Google (or Apple).
Would be nice to have local contact and calendar sync with my MacBook (OS X 10.9) but Apple made that hard, not the lack of apps on the phone so far as I can tell.
Rgds
Damon