Oliver Kamm provides the starting point: “religious faith is…a species of irrationalism.” This is not so much an insight as a tautology. Faith, by definition, is irrational.
However - and here Bunting is right and the new atheists mistaken - irrationality is a ubiquitous and in some ways desirable aspect of life.
It is irrationality - in the sense of over-confidence - that gives us so many artists, musicians, innovators and entrepreneurs, as a cool-headed rational appraisal of the chances of success would stop many from entering these fields.
Irrationality also underpins family life*. There’s nothing especially rational, from the point of view of the individual, about loving one’s parents or spouse. How many families are motivated to stay together by a simple error - the sunk cost fallacy?
And irrationality is also the foundation of politics. It’s trivial that instrumental rationality is no motive for voting. I’d go further, and suggest that our political views arise from accidents of our backgrounds or from tribal loyalties, and rationality - if it operates at all - mere gives them an ex post justification. Richard Dawkins himself has demonstrated this. At least one of his political opinions is founded on the groupthink that he despises in religious believers.
Now, the point about these irrationalities is that they help enrich our lives and sustain communities.
Irrationality, then, can have a valuable role.
So what is the offsetting virtue of rationality? Yes, it can be a way of finding the truth, or at least progress thereto. But it can also be an affected pose, a set of conjuring tricks with which we flatter our ego and impress the gullible; I owe my livelihood to this.
And rationality is not the only road to knowledge. When she says: “Certain forms of knowledge only come with practice” Karen Armstrong is echoing Michael Oakeshott, who argued that rationality was only one form of knowledge, and tradition and practice another. Which is surely right. You learn to cook, or play a musical instrument, or strip an engine, not by the application of abstract rationality but by practice. There’s lots of wisdom that does not arise from rationality.
All I’m saying here is that the statement “religion is irrational” is unhelpful. It’s true, but irrelevant. Instead, the questions are: is religion, like family life, one of those irrationalities that enriches our lives? Is it, like music or cookery, one of those wisdoms that come from tradition and practice rather than from rational calculation? Or is it instead a blight on our lives?
Insofar as the debate between the new atheists and religion has any meat, it is one of empirical sociology, not one about the nature of rationality.
*I'm using "rational" here in the sense of beliefs being proportionate to the evidence, rather than in the sense of what is good for us.