I begin with my thoughts, but feel free to skip those to get to the last part of the post where William Abraham has an interesting observation!
Perhaps we would make more progress if we recognised and delineated different forms of evangelicalism in our midst, and addressed how these differences affect the very process of trying to find some common ‘core’ or ‘centre’ around which we (might, hopefully will) unite.
Consider, for example, ‘Reformed evangelicalism’ (represented today in the Diocese of Sydney, in the Reform movement in England, and in the Church of England in South Africa) which not only draws on the Reformation (obviously) but leans heavily towards the theological commitments in the Reformation which led to the formation of ‘Reformed’ churches, distinguished on the European continent from Lutheran churches, and represented in Great Britain in the Presbyterian Church of Scotland and in the Puritan wing of the Church of England. Reformed evangelicalism is one form of conservative evangelicalism, but I do not think it is the only form of it. Another form of it sits more comfortably with Hooker’s critique of Puritanism, and with the historical development of Anglicanism which has re-embraced its Catholic (or, if you prefer, ‘catholic’ with a small ‘c’) heritage (that is, a re-formed-Catholic heritage). I am not sure if there is any particular label already applied to this form of evangelicalism (here I will call it Hookerian evangelicalism)*, but it is an important form since when we find dispute within the ranks of conservative evangelicals then the dispute is often between these two forms, probably to the bewilderment of participants who have not thought much about the significance of these differences. Another dispute within these ranks can arise between conservative evangelicals open to charismatic theology and practice and those critical of such theology and practice (here Hookerian evangelicals and Reformed evangelicals often find common cause).
Now, we can extend this line of thinking to name various other forms of Anglican evangelicalism and describe their points of difference. Causa brevitatis I will simply make an observation about Reformed and Hookerian evangelicalism (RE and HE respectively) – an observation, incidentally, which I am attempting to make as far as possible without endorsing who is right and who is wrong, because here I am interested in clarity (if possible) in understanding the road-blocks on the way to Anglican evangelical unity. Both forms of evangelicalism would emphasise their commitment to the authority of Scripture but in the context of the Anglican church, that is, current Anglican life historically grounded in the Reformation, RE understands the authority of Scripture (IMHO) in a way significantly different to HE. For RE ‘the authority of Scripture’ is a (or, the) supreme theological value so that everything is subject to critique by Scripture, including everything in the Anglican church. Effectively, nothing is sacred and nothing is immutable in the RE view of life, except for Scripture, its reading and its proclamation. The outstanding 2008 expression of this approach is the Diocese of Sydney agreeing to a resolution that means deacons may preside at the eucharist: Anglican orders, in the light of Scripture, are not immutable. By contrast HE understands the authority of Scripture in the historical context of the Anglican church: some matters of order and practice are immutable, their coherence with Scripture having been settled in the past. Clearly HE can be subject to sharp criticism from RE since HE decides on a case by case basis which matters are settled and which are not. “How come,” RE can say to HE, “priestly presidency is an immutable matter, but male priesthood is not”; for HE by and large has accepted female priests. Yet RE is not beyond criticism by HE: “How come RE wishes to remain Anglican while holding so few Anglican distinctives as immutable?” Not only is this significant difference, but it can be played up into a larger difference than it actually is, characterised in terms that appear to have a chasm between them: “Scripture” versus “Anglican”.
The reality is that both RE and HE are Scriptural and Anglican but the key to mutuality is understanding the special way in which RE is Anglican (through understanding that part of the English Reformation into which it is rooted and its deliberate distance from aspects of subsequent historical development of the Anglican church) and the special way in which HE is Scriptural (through understanding that changes it endorses (such as women’s ordination) are based on Scripture, and recognition that HE deliberately refrains from new Scriptural critique of various Anglican matters, on the basis that these are already established as coherent with Scripture.
All of which is to say, there is no substitute in any part of Anglicanism for respectful conversation: patiently exploring each other’s presuppositions, lovingly appreciating each other’s perspectives, carefully making proposals, and courageously receiving responses to those proposals.
[*'Hookerian evangelicalism' may seem a strange term, but we need something other than 'conservative evangelicalism' since, relative to Reformed evangelicalism, that term is imprecise, since RE is a form of conservative evangelicalism. 'Classical evangelicalism' is inappropriate since the current debate, in part, is a debate about whether RE or another is the true heir today to classical evangelicalism from the past; ditto 'mainstream evangelicalism'. A tempting contrast is 'Fulcrum or centrist or open evangelicalism', but Fulcrum in relation to its website is an interesting amalgam of conservative evangelicals, liberal evangelicals, open evangelicals, etc; 'centrist' in this context is much controverted, and 'open' appears to include evangelicals who can scarcely be described as 'conservative'].
William Abraham, in an essay 'Canonical Theism and Evangelicalism' in Canonical Theism: A Proposal for Theology and the Church (Eerdmans, 2008), after recalling the distinctive differences between the likes of Luther, Calvin, Edwards, Wesley, and Henry notes:
"evangelicalism is marked by an inner contrast and rivalry to bring to birth the heart of the Christian faith theologically and spiritually. Within the one family, the members have their own way of articulating its treasures and resources. This accounts for the internal tensions, the polemical edge, and the genuine differences of tone, practice, and content that are visible to the serious student."