My understanding is that, to oversimplify a little, Van Til and company were skeptical of the fact that Barth took the Christ-event, rather than the scriptures, to be the foundational element of his dogmatics. I suppose they thought that the former is mediated exclusively and only through the latter — so to know anything about Christ one must accept the truth of scripture.

I’m confused by your statement that Van Til &co engaged in “one-to-one relational evangelism with people who are sceptical about the whole basis of Christianity”. Surely the whole point of their system was that they did the exact opposite. Their primary assumption is that “the whole basis of Christianity” and the truth of the scriptures in particular has to be taken as axiomatic & that there is no neutral shared point between the believer and the nonbeliever from which an argument for the rationality of Christianity might be staged. Rather, the assumption is that the worldview of the nonbeliever must be inherently self-contradictory and far from trying to convince a person that the “basis of Christianity” is sound they would attempt to demonstrate that the basis of the unbeliever’s worldview is unsound. Schaeffer called it “taking off the roof” and advised it be done gently and with care for fear of bringing the other person’s world crashing down around them before they’re ready to accept an alternative. It usually devolves to somebody in the pub all frothy-mouthed desperately resorting to “so you’re saying there’s no basis for saying Hitler was wrong, is that it!?”

The irony is that the idea that there is no neutral staging ground for a rational demonstration of the truth of belief is one Barth clearly argues for in his polemic against natural theology. Indeed, your quote above seems like the sort of thing a reformed presuppositional apologist would embrace. Moreover, the idea that ultimately the only real polemic is that of God siding with the polemicist and revealing himself is the sort of thing a reformed thinker should cheer in agreement with.