We could go back even further and discuss the alliance of Church and State under Constantine and the development of Christendom, but let's not go there. The question is, 'Should the Church of England remain Established?' I would argue that it should not, because the idea of an established Church is alien to the New Testament. Under the old covenant there was no distinction between the religious and civil aspects of Israel's life. The nation was a theocracy - God's chosen nation, living under the terms of his covenant. But all that changed under the new covenant. Now the people of God are gathered from all nations. The Church may be a theocracy under the lordship of Christ, but she is distinct from the State. Church and State are two very different institutions. The State has been ordained by God to restrain evil and preserve peace and order in society (Romans 13:1-7). But the Church has been called to carry out her Great Commission to preach the gospel and make disciples for Christ from all peoples. The State may use force to subdue law breakers and protect its citizens. The Church's only weapon is the sword of the Spirit, the word of God. State establishment obscures the Church's unique gospel-centred mission. That is why there is no sense in the New Testament that the Church should aspire to establishment by the State. Obviously, that kind of thing would have been impossible anyway under Nero. But the apostles don't so much as hint that establishment would be in any way desirable. All they asked was that the State tolerated the existence and activities of the Church (see Paul in Acts). The apostles would certainly have been outraged at the thought that the State should appoint Church leaders. However the Church/State distinction found so clearly in the New Testament was gradually eroded away from Constantine onwards.
Even the Reformers were willing to use the powers of the State to further their cause. They are called Magisterial Reformers because they expected the Magistrate to help reform both church and society. In 16th century England, some Protestant got so fed up with the slow pace of Reform in the Church of England, that they took the radical step of separating from the established Church. In the words of a title of one of their books, they believed in Reformation without Tarrying for Any. These Separatists, men like Henry Barrowe, John Greenwood and John Penry argued that the Church should not have to wait for permission from the State to implement reform. This was seen as so subversive of the unity of the country that some Separatists were actually put to death. But it slowly began to dawn on more and more Protestants that Church and State should be separated. The Independent Puritans tended to this view, while Presbyterians held that the godly Magistrate had a duty to assist with Church reformation. The 1689 Baptist Confession amends the Westminster Confession's section on the Civil Magistrate(here), to limit the State's role in Church affairs (here).
Some would like to see the Church of England disestablished for secular reasons. They resent the intrusion of Christianity into public life. But that is certainly not my motivation. Christ's lordship is not limited to the Church. He is Lord of all. Christians should act as salt and light to influence the direction of their country. We can do that by scrutinising legislation, writing to M.P's, lobbying Government ministers and so on. It would be a good thing if more genuine believers entered politics to bring Christian values to bear upon the public square. The fact that the Constitution of the USA forbids the establishment of a Church, does not mean that Christianity has no voice in public life over there. In fact it is a strange paradox that in England, with its officially established Church, Christianity is often banished to the sidelines. In 1914, the Church of England was disestablished in Wales, largely due to pressure from the Nonconformist Churches. Isn't it about time that England got up to speed?