Earlier quoted context omitted.
Sure, there's a strong public interest in having proceedings on record. US civil cases are supposed to have a presumption of openness, which the judge weighs against other interests, like protecting trade secrets, confidential business information, privacy of third parties, etc. The public record argument is fine; it's just a different argument than the extrajudicial punishment advocated by the original post.
I think the extrajudicial punishment he's advocating for is the wrath of the court of public opinion though? Unless I'm misreading.
Suppose the government charges you with murder, searches your house, and finds your sex toy collection. At trial they present some elaborate thesis about how you used a sex toy to kill someone, but do not convince the jury, so you're found not guilty. The public has a legitimate interest in judging that the trial was handled with integrity and that the correct verdict was reached. They do not have a legitimate interest in judging you based on whatever private information presented at trial might in some way embarrass you (eg, photos of your sex toy collection). On balance, it could be that the public-record interest does in fact justify making public the evidence of the sex toys, but you have to justify it on those terms. The transparency is not itself intended to be punitive.