Does this quote from Origen prove the Bible is corrupt?

The quote proves that Origen believed it was not possible to discern the true original text, because copyists--either through negligence or "audacity"--had failed to preserve the original with due diligence. Modern scholars, including those cited above, often share this attitude. Ehrman in particular is hardly confident that original text of the NT can be reconstructed, even though he does say that most verses can.

What good is it to say that the autographs (i.e., the originals) were inspired? We don't have the originals! We have only error-ridden copies, and the vast majority of these are centuries removed from the originals and different from them . . . in thousands of ways.

The issue of whether the Bible as we now have it is "corrupt" touches a matter of faith for many readers. But for Origen's faith, this was not a challenge, nor does it need to be for modern Christians. God speaks to us through it, whether or not every jot and tittle matches the first written version.

"Corruption" is a term used by textual critics to refer to any modification of the original text, whether intentional or accidental. Your focus should be to rightly define "corruption". The top NT scholars including Bart Ehrman believe that despite the minor textual corruption or interpolation, the NT books remain incredibly accurate to the original writings. It is a fact that some NT books were lost, and many of the OT books have been lost, and the fact of textual evolution or changes have been well attested. So, eventually what the "Christians" need to adjust is their perception of the nature of scriptures and inspiration. The vast majority of textual corruption had already occurred till the second century. Despite, the fact that some books may have been pseudonyms and full of midrashic creative revision or embellishment rather than accurate history. The historical value of the bible is not destroyed, but only the naive notions and dogmas of the devout man about the scripture.

In The Text of the New Testament its transmission, corruption, and restoration, by Ehrman and Metzger, pp 24, we read:

In the earlier ages of the Church, biblical manuscripts were produced by individual Christians who wished to provide for themselves or for local congregations copies of one or more books of the New Testament. Because the number of Christians increased rapidly during the first centuries, many additional copies of the Scriptures were sought by new converts and new churches. As a result, the speed of production sometimes outran the accuracy of execution. Furthermore, in preparing translations or versions for persons who knew no Greek, it occurred more than once (as Augustine complained) that "anyone who happened to gain possession of a Greek manuscript and who imagined that he had some facility in both Latin and Greek, however slight that might be, dared to make a transladon" (De doctrina Christiana, n.xi.l6).

The book testifies that despite the textual corruption, the NT is highly reliable to the original:

In contrast with these figures [about non-Christian Roman writers], the textual critic of the New Testament is embarrassed by the wealth of material. Furthermore, the work of many ancient authors has been preserved only in manuscripts that date from the Middle Ages (sometimes the late Middle Ages), far removed from the time at which they lived and wrote. On the contrary, the time between the composition of the books of the New Testament and the earliest extant [existing] copies is relatively brief . . . several papyrus manuscripts of portions of the New Testament are extant that were copied within a century or so after the composition of the original documents. (Metzger and Ehrman, p. 51)

The scribes were often amateur and unlearned on language and religion, as Augustine himself was, it is not surprising to see the quote of Origen about the early corruption, but we need to be careful to understand they meant it concerning the minor interpolations and errors which do not make a big change to theology. The corruption can be sorted out and studied under textual criticism. The reason why modern man is baffled by the textual changes or corruption is due to his comic and limited perception of the ancient writings & religion.

No, the quote does not "prove" that the Bible is corrupt. This is clear from the very book that you're quoting; Metzger and Ehrman note that Origen often based such claims "on considerations other than those of a purely textual nature" (2004, p. 201). Looking at the fuller context of the passage you quote, Metzger and Ehrman note that "because of some exegetical difficulty, Origen suggests that perhaps all of the manuscripts existing in his day may have become corrupt" (ibid,. p. 201, emphasis mine). In other words, Origen made this suggestion on theological and exegetical grounds, not textual ones; the manuscripts seemed to go against his preferred reading, and so he suggested that they had been corrupted. This does not in any way demonstrate that the Biblical texts actually were corrupted.

On a broader note, the vast majority of scholars do not believe that the Bible is "corrupt." As Bart Ehrman (a famously skeptical NT scholar) observes, "scholars are convinced that we can reconstruct the original words of the New Testament with reasonable (though probably not 100 percent) accuracy" (2000, p. 443). While there is certainly disagreement about the exact reading of particular verses, it remains the case that "In most instances, scholars are unified in thinking that the author probably wrote a verse in this or that way" (Ehrman 2017, p. 11).