This. A thousand times this.
Bias is an inherent part of language, and we cannot eliminate it, because bias is inherent in both what is said and what isn't said.
Even beyond a simple textual level, you have to arrange content somehow, and that in and of itself is subject to bias. What do you put linked to direct from the home page? Do you entirely forego headlines? If so, what do your links on your home page say? All of that is influential in how someone perceives what you've written.
If we take Bannon's appointment to Trump's administration (and I'll preface this by saying I'm not American, and there's a limit to how much attention I pay), we can easily write headlines like:
To start with, some real ones:
* 2 Trump Picks: an Insider and a Firebrand (NYT, 14 Nov) — the former is either a positive or negative depending on your political leanings, the latter I don't know whether it's connotatively good or bad in en-us
* Trump Draws Criticism for Far-Right Pick (NYT, 14 Nov) — entirely factually accurate (so… not bias?) but omits any negativity in mentioning as to why he's drawing criticism (so… bias?)
* Trump makes controversial pick for senior aide (AP, 14 Nov) — ditto
* Trump Taps Reince Priebus as Chief of Staff, Steve Bannon as Chief Strategist (NPR, 14 Nov) — entirely factually accurate (so… not bias?)
Some hypothetical ones:
* Trump names nationalist as senior aide — I think there's consensus across the board that that would be accurate, but whether it's positive or negative depends on the reader, so is arguably bias
* Trump picks fringe right winger as Chief Strategist — disassociates him with most of the right wing, mostly negative connotation, obviously bias
* Trump Choses Racist — this is obviously an extreme example, but it's worth looking at: is it impartial? if not, it's obviously bias. do we have direct evidence that Bannon is racist (v. being on the board of a newspaper publishing such content)? if not, it's likely not entirely factual.
Obviously, we can debate how, and in what way, each of those is bias, but they all lead the reader to view Bannon in a specific way before the article is ever read (except, perhaps, the NPR one).