I accept that Barth is important for the mainline Christian tradition, but I have found his incredible verbosity and vagueness sufficiently hard to deal with that I’m not directly influenced by him. I tend to read Biblical scholarship and more recent theology. For a question like this I’d look at PCUSA theologians, particularly Gerrish and Ottati.
Both of them look like paleo-Protestant in your sense. Their works start by reviewing the tradition. To respond to this question I reviewed Gerrish’s book “Saving and Secular Faith.” He reviews Luther, Aquinas and Calvin on the nature of faith, and adopts a somewhat modified version of Calvin’s definition. He sees faith as resulting from the impact of the Gospel, which for him is the way grace functions. Ottati’s works follow a similar approach of starting with a review of the tradition. He is more explicitly an advocate of the primacy of grace. He sees himself as part of the Augustinian tradition, through Calvin.
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On Barth: I think the modern mainline tradition is part of a historical development of Protestant thought going back to Luther and Calvin through people such as Schiermacher and Ritschl. Barth explicitly divorced himself, not from church tradition as a whole, but from the liberal tradition. I suspect this was because he blamed liberal Protestants in Germany for giving in to Hitler, and took that to be due a problem with the liberal tradition as a whole. That cut him off from a large part of 19th and 20th Cent Protestant history. I don't know whether his judgement was right in Germany, but in the US, liberal Protestantism has generally been quite willing to criticize the culture and government actions. But Rauschenbusch and others critical of the culture were influential in the US, but possibly not in Germany.
