Here is Alan Hirsch’s main concern for Calvinism (he said it in the comment section):
My main concern for Calvinism of any variety (and I was trained within the tradition) relates to its…
1. Spiritual disposition: There seems to be historic propensity for religiosity and harsh judgmentalism. I can’t see much change here in the new/neo Calvinist/Reformed expressions so far. Tim Keller being the major exception as far as I can see.
2. Ecclesiology: The ecclesiology is hardwired into the theology. It seems to lack the agility needed to be fully responsive missionally. Geneva still holds its imagination. In other words, its ecclesiology precedes and forms its missiology.
First, I think that there is harsh judgmentalism and religiosity among Calvinists. I know I certainly struggled with it much in the past and still do at times today when I encounter Christians from other traditions. But I do think that a true understanding of Calvinism (by which I mean a narrow – sovereignty of God in salvation as ultimately responsible, the inability fo man, God’s free choosing, God drawing sinners, and God preserving all of those he’s drawn) should promote humility. There are more examples of exceptions than Tim Keller. John Piper, D.A. Carson, Mark Dever, Philip Ryken, Ligon Duncan, C.J. Mahaney are a few that come to mind (if you want a solid understanding of Calvinism that humbles you to the dust and exhorts to continue in such humility, listen/watch these). But then again, I heard in a audio lecture, Hirsch said that he said he shouldn’t be so sure about something to preserve some humility. So maybe his definition of humility being unsure about a particular concept or assertion being true is humility. To that I can respond no better than G.K. Chesterton who said:
What we suffer from today is humility in the wrong place. Modesty has moved from the organ of ambition. Modesty has settled upon the organ of conviction; where it was never meant to be. A man was meant to be doubtful about himself, but undoubting about the truth; this has been exactly reversed. Nowadays the part of a man that a man does assert is exactly the part he ought not to assert–himself. The part he doubts is exactly the part he ought not to doubt – the Divine Reason. . . . The new skeptic is so humble that he doubts if he can even learn. . . . There is a real humility typical of our time; but it so happens that it’s practically a more poisonous humility than the wildest prostrations of the ascetic. . . . The old humility made a man doubtful about his efforts, which might make him work harder. But the new humility makes a man doubtful about his aims, which makes him stop working altogether. . . . We are on the road to producing a race of man too mentally modest to believe in the multiplication table (G.K. Chesterton, Orthodoxy [Garden City, NY: Doubleday and Co., 1957], pp. 31-32).
Secondly, if you take a broader view of Calvinism which includes Calvin’s ecclesiology then yes, Hirsch is correct, the ecclesiology is hardwired into their Calvinism/Reformed understanding. But if you define Calvinism in the narrow way as the doctrines of God’s grace saving man despite his rebellion and initial unwillingness, then the ecclesiology is not hardwired into their Calvinism. So the Presbyterians’ ecclesiology is intertwined with their Calvinism (I think, someone correct me if that is wrong). But Baptists, free church, congregationalists, Anglicans and Episcopalians, and Methodists (Whitefield was one) all have ecclesiologies that are not tied to their Calvinism. Furthermore, there are missional Calvinists who’s ecclesiology would be commended by Hirsch (like Tim Chester, Steve Timmis, The Crowded House, Soma, Kaleo). They scatter churches and reproduce in ways that are in line in large measure with Hirsch’s thinking (ecclesiologically), yet they are solidly Calvinistic (in the soteriological sense).