This idea really isn't that original. Philosopher, journalists, and others have done this for quite some time, and this has, I'm confident, had significant positive effects on the public debate. Religious people were forced to improve their arguments as a result of philosophical criticism from Hume and others, for instance. Also, even if you might think that the political debate is bad, imagine how bad it would have been if there were no journalists reporting relatively objectively on politics.

My suggestion is thus just to make existing efforts at objective reporting and criticism more systematic and comprehensive: it's not a qualitative leap, but just a quantitative shift. I don't see why that would be impossible in principle (though it may be hard in practice) or why that would not have any effect, given how much our present institutions for objective criticism actually has achieved (in my view).