He (rightly) bemoans the "debased politics" of hostility to migrants and benefit claimants, but seems to think the answer is a "People's parliament", in which parliament is more open to meetings of ordinary" folk.
This misses something - that the public supports caps (pdf) on migration and welfare spending; for example, in a recent YouGov poll (pdf), only 15% said welfare benefits should be protected from further cuts.
"Debased politics", then, doesn't represent a failure of democracy but rather the triumph of it. The problem with politics is not that politicians are out of touch with voters - but that they are too much in touch.
And this is where cognitive dissonance enters. Some of the left can't see that their cherished beliefs - in democracy and in income equality and tolerance - are incongruent. Rather than acknowledge this conflict, people like Mr McDonnell seem to want to try and dodge it, by hoping that a different form of democracy will somehow produce healthier attitudes.
This is theoretically possible; maybe some forms of deliberative democracy will cause prejudice to be eroded by contact with facts - but this is not assured, and Mr McDonnell gives us no reason to suppose it will happen. Indeed, research suggests that exposure to evidence that opposes our beliefs can merely cause us to entrench (pdf) our beliefs still further.
What the non-Marxian left is missing here is that capitalism sustains itself in part by generating cognitive biases - ideology - which serve to support the existing order and oppose egalitarian policies. In other words, there is a conflict between (capitalist?) democracy and justice.
Although this seems - and is - Marxian, it leads to a conclusion most associated with Isaiah Berlin:
Some among the great goods cannot live together. That is a conceptual truth. We are doomed to choose, and every choice may entail an irreparable loss. (“The pursuit of the ideal", p11 in The Proper Study of Mankind)