The Right Wing (in the US, our Republican Party) is less identifiable by its right-wing policies as it is by its right-wing epistemology: they reject all scientific evidence for climate change, biological and geological history, medicine, etc. For the religious, this manifests itself in the old epistemology that says the Bible is the sole source of all Truth. For the non-religious, this tends to manifest itself as total nihilism, which is expressed in the more heartless strains of Libertarian Objectivism; for these people, might equals right and society is a zero-sum game where the only truth is winning and losing.
And this epistemological divide is the one that is tearing us apart. When you say there's no serious right-leaning counter-weight to (the NYT et al.), what you mean is that there is no serious news outlet that doesn't hold the Enlightenment values. And when you put it in this light, it's an obvious statement, isn't it? Politically speaking, the WSJ and the Economist can very much be considered right of center (leaving aside that the median center has moved right over the last 40 years), but they are still rooted in Enlightenment science and rationalism.
Point is, although the political right wing is currently the main base of the epistemological right wing, it doesn't necessarily have to be that way. And the problem with saying we need more right-wing voices in mainstream media is that there are so few of them who meet the basic epistemological requirements; when they do, they are often rejected by the broader right wing for being too far left!