That’s right, I use F-Droid in preference to Google’s Play store wherever possible. In fact I’ve taken to running my Android phone logged out of all Google services.
Of course this isn’t how Google designed things, but given the choice I’ve come to use alternatives to their services for most everything I do. My calendar, contacts, bookmarks and shared files are mostly available from a personal ownCloud instance on the Internet. F-Droid actually has most of the software I’d need or want. That includes Firefox, K-9 Mail, C-SIP Simple and others. I’ve also pulled some packages off Amazon’s app store, like ES File Explorer (still looking for a decent open source file manager that includes network transfer but doesn’t require root access).
It’s not that I don’t trust Google. Well, actually it is.
A better option would be for me to run CyanogenMod or Replicant, as I do on one of our Samsung tablets, but neither of those projects is interested in the relatively inexpensive smartphone model I own and so there aren’t any reliable images for it. As a result I’ve continued to run stock AT&T Gingerbread far longer than reasonable, but such compromises are a part of life we sometimes have to work with.