As someone who runs a niche news site, it's hard to see the point focusing on running a Facebook page anymore. We kind of go through the motions to keep up with appearances, but Facebook is less than 5% of our total overall traffic, and 95% of that portion is all from individuals sharing our content, not from our page directly. It's just impossible to "bid" on traffic via Facebook and win out over companies that have…
Just because your FB shares don't directly feed your traffic doesn't mean they're useless. You might try A/B tracking on promoting stories through FB or not, and see how they track for views.
(Though I've also had experiences with organic spread of stories -- sometimes something just hits a nerve and balloons.)