As for Methodism, it, too, began as a reform movement within the Church of England, emphasizing moral reform and holy living. The Wesleys did not deal with civil matters, but emphasized the charity of Christians as the means of assisting the poor. The Salvation Army, an offshoot of Methodism, best illustrates the Wesleyan approach to social reform: small scale and person to person, the antithesis of the modern welfare state. That the Northern and later the United Methodists of the 20th Century became an "Amen corner" for big government is reflective of the rise of liberalism in that denomination. The "Social Gospel" preached by men such as Walter Rauschenbusch and Bromley Oxnam was merely humanistic socialism with a Christian veneer. Rauschenbusch and Oxnam were heretical in other respects, such as denying the plenary inspiration of Scripture.
There was aspects of these three Calvinist states that were undesirable: intrusion into private and family affairs by civil magistrates to ensure church attendance and punishment of those who did not hold to church doctrine. However, these intrusions were known throughout Europe, in both Catholic and Protestant states, as seen by the Inquisition in the Catholic states. The Reformers were not consistent in their application of Scripture to civil government and relied instead on medieval legal theories that did not distinguish between the proper and separate roles of church and state. The full development of the concepts of freedom of conscience and limited government occured after the 17th century wars of religion. This development was found in the writings of Calvinists (Samuel Rutherford, John Witherspoon), Anglicans (Blackstone), nominal Christians (John Locke), deists (Jefferson, Montesquieu), and Catholics (Lord Acton, Frederic Bastiat).
That being said, there was a strain of Protestant social uplift and reform that derived from the Second Great Awakening (circa 1800) and particularly the Third Great Awakening (circa 1850s). These revivals brought the Baptists into prominence as well as new groups such as the Campbellites (Churches of Christ) and the precursors of the Pentecostal movement. The theology preached by such revivialists as Dwight Moody, while orthodox in many respects, rejected the Reformation concepts of the bondage of the will and the sovereignty of God. Instead, emphasis was placed on the role of the individual will in accepting Jesus as Savior, as well as the ability of a saved person to lose his salvation through sin. Because of the emphasis on individual choice and a lack of reliance on God's sovereign grace, these Christians started looking to government as a means of preventing people from going astray.
Thus, we see the rise after 1880, particularly in the Midwest and South, of the Populist movement, of which Prohibition advocacy was an aspect. Populists favored government regulations on railroads, price supports for farmers, and a inflation of the money supply. They also favored alcohol and later drug prohibition and strict anti-pornography laws. While the Populists were, by and large, professing Christians, they rejected the reliance on God's grace and sovereignty that characterized Calvinism. This doctrinal flaw made them susceptible to the appeals of non-Christians and nominal Christians who saw in the state a means of creating a "heaven on earth."
Populism a "red zone" political movement, united with the Progressives, who were the heirs to the Unitarian based social uplift movement that began in the Northeast before the Civil War. In essence, Progressives such as Woodrow Wilson told their Populist counterparts such as William Jennings Bryan, "Sure we'll agree to Prohibition, but you must support an income tax, direct election of Senators, and a Federal Reserve System in return." This alliance, which by the era of the New Deal was joined by the rapidly rising union movement and a majority of white Catholics, was responsible for turning the Democratic Party of Jefferson and Jackson into the prime advocate of expansive Federal government power. But in this alliance, it was the humanists who were in the driver's seat, and the revivalists and later the Catholics who were along for the ride.
Historic Protestantism, whether we look at the 16th and 17th Century Reformers, or the later Pietist and Methodist movements, placed a strong emphasis on moral living and personal responsibility for oneself and for others. The primary thrust for social reform through government action derived from Unitarians and others that rejected much of both the core of Christian doctrine and the Reformation distinctives. That there were evangelical allies to the social reform movements after 1880 cannot be denied. However, they were not in charge of the overall thrust away from inedividual responsibility and toward statism. Also, these evangelicals rejected several key points of Calvinist doctrine, notably the bondage of the will and the sovereignty of God. To characterize these evangelicals as heirs of the Puritans or the Scottish Covenanters is to do violence to both history and theology.
While Unitarianism may have been behind many of the more radical mid-nineteenth century reformist trends, the desire for change also made its way through other traditions, as you note, so I don't think I don't think it valid to link reform movements and Unitarianism as tightly as some do. Unitarians may have been a vanguard, but those Methodist reformers and populist evangelicals you site didn't need to take orders from them. Remove the Unitarians and the flavor of politics and intellectual life would have changed, but the actual policies might not have been so different.
To characterize these evangelicals as heirs of the Puritans or the Scottish Covenanters is to do violence to both history and theology.
That is certainly a very Calvinist insistence on clear lines and distinctions. But sometimes ancestries aren't so clear and unambiguous. Confronted with social conditions that they disapproved of Calvinists in various countries did come to adopt social and political ideas that led to statist policies. Is this heresy or an ongoing evolution sparked by the interaction of religious ideas and social conditions? Some of the reconciliations of Calvinist theology and state action were quite ingenious and sophisticated.
I would guess that where we differ is that you are looking at a clash of religious ideas, and I am looking at how interests, desires, and ideas make their way felt in religion and politics. So rather than see an ongoing battle of theologies, I see ideas and conditions affecting various religious traditions in similar directions. Seen from a strict Calvinist perspective, every step away from Calvinism probably does lead inevitably and ineluctably to our present day troubles, but that's not my perspective. American religion, like American politics has long been a kind of halfway house, but that condition hasn't been a complete loss.