I have witnessed the legacy of uniformity in the pre-eminence of Anglican religious influence on the political establishment, not only in the UK, but also several territories in the West Indies, where I and my forbears previously lived. Typically, you could not even hope for a job without a character reference from the Anglican priest. In spite of this requirement, the lives of many of the established clergy (who carefully restricted references to Anglicans in good standing) were far from exemplary.
Beyond this anecdotal evidence, many studies of other parts of the former British Empire attest to the sway held by Anglicanism over the colonies. Perhaps, the post-colonial experience in the USA differs from the West Indies. Factually, we do know that it was the political influence of key members of the Church of England that established several colonial authorities, such as Virginia and Georgia (e.g. cf. Virginians Reborn: Anglican Monopoly, Evangelical Dissent, and the Rise of the Baptists in the Late Eighteenth Century). These colonies were committed to the predominance of Anglican influence through a provincial cultural hegemony: a mix of compulsory catechism, conspicuous churchmanship and insistence on externalised form and ritual. It is the insidious legacy of this kind of provincial uniformity that I consider to be malign and ripe for dismantling. Human beings may be won to a cause by the tyranny of the majority. They will desert it with contempt, once the old regime of coercive tactics is exposed. Persuasion of the truth is a far better way for Church of England to secure a lasting posterity.
We may note the greater popularity of other denominations in the US, such as the Baptist movement. However, their later ascendancy was largely a result to the American Revolution that created a new environment in which evangelical Protestants of other denominations could flourish.
While, in each case cited, the legal framework of uniformity (upon which the moral case for colonialism was built) was repealed, the history of how provincial uniformity was imposed continues to influence how Western ideals are established in the world and elsewhere maintains the perception of Western liberal thought as another form of neo-colonialism.
The reaction of conservatives is ironic, given that the Covenant reduces their own autonomy, rather than maintaining the relative freedom of the status quo.
While provinciality may indeed be a worthy ideal for internal self-government, the historic role and influence of the Anglican church is far from ideal. Especially in former colonies, it is tarnished with the brute force of colonial rule and, rightly or wrongly, viewed with suspicion of surreptitiously imposing its values on the wider society.
If there is a demonstrable commitment to undo the entire legacy of uniformity, it may engender enough trust between provinces for us to settle or accept differences without recourse to a written covenant.