You are wrong to say that the quote implies that the authenticity and authority of Scripture cannot be argued for reasonably. It’s somewhat funny to accuse Clark of abandoning logic, as it is his use of logic that leads so many people to object to him. But perhaps what you mean by “reasonably” is not “logical” but “starting from autonomous man’s authority,” as is commonly done? Clark does not abandon the use of reasonable argument, rather he rightly points out that there is no higher authority by which Scripture can be subjected to for confirmation.

As for your preference for building a natural theology – have you read Robert Reymond on the topic? I haven’t read his “Justification of Knowledge” yet but I would like to. Here are a few quotes from his Systematic Theology:

There is a legitimate sense in which the awareness of God that all people have by virtue of their being created in his image and by virtue of his inescapable revelation of himself to them in nature (Rom. 1:20) may be called “natural theology.” With this use of the term I have no problem: indeed, I wholeheartedly endorse it. But when I refer to “methodological natural theology” I am referring to the theological method whereby a “first floor” philosophical prolegomenon is first built by natural reason working independently with what is portrayed as “neutral data” upon which a “second floor” set of beliefs derived from special revelation is later placed. In this kind of “natural theology,” the Christian revelation, not intended to displace or to function as the ground of the philosophical prolegomenon, presupposes the philosophical prolegomenon and presumably confirms and supplements it. I argue against “natural theology” in this latter sense in “The Justification of Knowledge” (fn 13, p 137)

Much more could be said about methodological natural theology, but suffice it here to say that nowhere does the Bible endorse the notion that general revelation was given to provide people the data by which they might, beginning from themselves, reason their way to God. The Bible introduces general revelation alongside special revelation to emphasize man’s guilt. The entire effort of Thomistic natural theology to discover God by natural reason apart from Jesus Christ must be judged not only a failure (see 1 Cor. 1:20–21) but also as an unwitting handmaid of the entire revolt of human philosophy against the necessity of special revelation. See Robert L. Reymond, The Justification of Knowledge (Phillipsburg, N.J.: Presbyterian and Reformed, 1984), 118–30 (p. 57)

Finally, methodological natural theology does not square with the natural apologetic activity of the early church as we find it depicted in the book of Acts. The natural theologian maintains that it is not right to ask skeptics to believe in Christ on the basis of scriptural authority before they have had a chance to consider the evidence supportive of the Christian claims. But does the unbeliever possess some independent criterion of verification which can and should authenticate the truth of Christian revelation in advance of faith? I think not. Otherwise, we must conclude that Dionysius the Areopagite, who believed in Christ simply on the basis of Paul’s testimony prior to any investigation into what Paul proclaimed, was the biggest fool on Mars’ Hill that day in A.D. 50 (Acts 17:22-34), and that the most intelligent men there were those who determined to hear Paul again on some subsequent occasion! No, the missionary efforts of Peter, Stephen, Philip and Paul never urge lost men to do anything other than to repent of sin and bow in faith before Jesus Christ. When they debate, they draw their arguments from the Scriptures (Acts 17:2; 18:28)… (p 150)