Much of it pivots around George Lindbeck's view of doctrine, though I would tackle Lindbeck from a different approach - that of the modern theologians that led on to postmodern (specifically postliberal) theologians.
If you go back to Kark Barth, the God of revelation becomes so one way (from) and so high and dry, neo-Calvinist revelation and salvation, that if you are not one of the chosen you may as well be an atheist. Nothing of culture, or of human made religion, is of God: God chooses whom he contacts and where he encounters. He, of course, was involved slicing into history at the Gospel, so it is claimed, but even that cannot be made into an artefact of religiousness and is culture free. What this means is that there is no cultural earthly place in which to root objectivity. You could do it in God, but that cannot be from humans up, only God down, so that act of making objective is a false human initiative.
There is a direct connection from this to the postmodern, in that with no rooting in culture, culture is then just shifting forms. Hans Frei thus followed on from Barth, and the Bible is a dramatic encounter that you just do, without objectivity beyond its drama, and Lindbeck is the Church and doctrine equivalent. He wrote: (1984) The Nature of Doctrine: Religion and Theology in a Postliberal Age, John Knox Press. In it he rejects that cognitive which becomes the truth, because there is no objectivity, and he also rejects the primacy of experience, which is liberal and individualist. So you take something like ecumenical doctrine, and hold it without objectivity, and just perform.
The criticism is not that this is cultural-linguistic, but that it is frozen and becomes artificial. It takes what was cultural, at some point that is in history (so does not escape the cultural reference), and then says this must be the source of identity of community while it performs the basis of doctrine. The freezing lifts it out of history in any conencted sense.
Of course this is not rooted in anything, not even in storytelling, one to another, which can be just as relative, and so this "Yale postliberalism" has a kind of ice death about it. The question is why take this or that version of Christianity, because the selection and freezing simply reveals the changes that have taken place in time regarding belief and assumptions about truth. The use of a level of ecumenical agreement may just be a lowest common denominator, and that churches themselves usually express far more.
Lindbeck's scheme is called "postliberal", but then it should incorporate something of the liberal about it. It does not - it rejects liberalism. A far better cultural and linguistic approach focuses more on ethics and the relativity of faith in culture, and thus contains something of the liberal about it still (just as postmodernity has modernity within). This is Daniel Liechty's (1990) Theology in Postliberal Perspective, SCM Press, which I know from direct contact had earlier origins than Lindbeck's book, that overtook Liechty's for position with Yale University behind it.
One chapter of Liechty's book is 'Church without Dogma' (59-72) that concludes:
The emphasis on unity based on shared ethical commitment is one way to keep people in dialogue as they attempt to give expression to their spiritual experience of the world as trustworthy and loving. To give up the "authority" of a creed or a set of doctrines is indeed a risk. But it is a risk worth taking. (72)
This, then, is still open to experience and to linguistic formulation and to dialogue. It sees doctrines as artificial. It is still Christological, and again it is worth quoting from the final paragraph of the previous chapter, 'Toward a critical Christology' (50-58) where sympathy of communities' deification of Jesus over time does not mean uncritical acceptance, and Incarnation points towards historical relativity of even our notions of God (54):
What we affirm about Christ is inseparable from what we affirm about God and, ultimately, about our own lives. If grasping that present-future Kingdom of love and freedom is still an option for us, fidelity and Chritian faith obliges us to reach out for it, even if that means demythologizing and even abandoning entirely pre-existent/ incarnate/ exalted Christological language. After all, that is what Jesus would have done. (58)
Why would he have done this? Well Jesus pointed to God not himself (51). There are a number of reasons given why we would end a belief in a deified Christ, in that a deified Christ can leave to passivity, oppression, conquering, division from others in other religions (54), and, relevant to day's debate:
In as much as a deified Christ confirms us in our prejudices of race, religion, class, sex, and sexual orientation, we must be willing to let that deified Christ go. We must take a stand of unbelief towards that Christ. (54)
The point about doctrine then, in this postliberal view, is that it is subordinate to ethics and is just a set of areas to signpost and question.
John Watson is stronger about doctrine than this, but it is also there to provoke and be dynamic, to open out not close down, with much that is paradoxical. A vision of what is different from what we have does not support a closing into neat concepts. We should move to the relational aspects of faith, not simply cognitive.
Such is fine, but it does mean doctrine always has something of the provisional about it. The Trinity is just a metaphor, either of something stable (God, Being, ultimate concern) or in a quest of forever chasing other metaphors (depth, what matters, one's focus, an ideal...). My view is that these pointers are bound to have some role in a liturgical setting, though they are not exclusive, to make a pathway, for this relational purpose - but that there is little place else for confession of beliefs. Experience remains important, as does the far broader and wider investigative area of truths, even where drawing on the inheritance of traditional concepts is used to aid the spiritual path.