It's worth noting that Chesterton is a highly idiosyncratic author, and one who frequently portrays himself as out of step with his time. It may be more revealing to look at the way in which Chesterton himself was inclined to use the word "pink". In particular, I found a passage on page 448 from The Collected Works of G. K. Chesterton: The Illustrated London News (using Google Book Search) which runs:
though I am no admirer of Bolshevism, I am still less an admirer of pink. Pink seems to me an essentially false and negative colour; because it is the dilution of something that is rich and glowing or nothing. ... [P]ink suggests nothing but the horrible and blasphemous idea of wine with too much water in it. Pink is the withering of the rose and the fading of the fire pink is mere anemia in the blood of the universe. And there is a merely pink humanitarianism which I dislike even more than real Red Communism. It is not so honest; it is not so genuinely angry or so justly angry; and it is ultimately every bit as negative and destructive of the strong colours and definitive shapes of any great historical culture. ... This cold and colourless sentimentalism ... threatens the world like a slow and crawling Deluge
which may shed some light on "the pinkest of papers"
There is always a risk, of course, in attempting to interpret an author's fictional work in light of work that was written as non-fiction. But given that Chesterton is, in my opinion, an obnoxiously mono-vocal author, that, in many of the stories, Father Brown seems to speak Chesterton's mind, and Father Brown's overall reaction ("I cannot think at this moment of anything in this world that would interest me less. And, unless the just anger of the Republic is at last going to electrocute journalists for writing like that, I don't quite see why it should interest you either.")