Lots of them have. Look at any site where science skepticism is regularly posted, you will find that maybe a good 50% of the content is commentary on papers. Of those who don't trust science and haven't read papers, they will usually have read commentary by those who have.

Source: I've published such skepticism in the past and met people who read my articles, including politicians.

Nonetheless, you're right that simply publishing peer reviews won't help improve trust. The situation is bad enough that there's no One Weird Trick that can dig academia out of its hole. Some of the most intellectually fraudulent papers I've written about in the past did have transparent peer review, and all it showed was that peer reviewers were often aware of the critical problems found inside and waved it through anyway. Or that their feedback was ignored. Or that they agreed with obviously bogus practices. Or that the parts of the paper that revealed the problems weren't reviewed (eg. appendices, github repositories). After all, nobody cares about papers that got rejected by the system, the distrust is driven by the papers that weren't, so almost by definition such papers either were badly reviewed or the review wasn't used.

I think there are a few groups and reasons of distrust. Some more or less valid.

Those that distrust authority as a whole and lean into conspiracies cannot be saved with this kind of thing.

But i think news and science are having similar perception issues recently.

Distrust for news growing among the average population (for some good reasons). People are loosing faith in the objectivity of established media organisations. Most people are exposed to science through these traditional news.

So adding back some sense of confidence and authority to scientific institutions is very valuable to non-academics. Even if they themselves would not read the papers or revews.

>Of the people who distrust science,

Why should anyone trust science? Skepticism should always be the default position. Putting it on a pedestal to be worshipped is what led us all into this mess. If science needs trust to work, then whatever it is doing is something I'd like to see fail.

I would go further: Anyone who has published a "scientific paper" in the last decade or so either "distrusts science" or is more likely than not a mid-wit at best.

Your posting doesn't give me the impression that you're very familiar with "science".

The fact remains that distrust of science continues to grow. Up to now the establishment’s response has been one of condescension. Your comment echoes that attitude.

Ignoring the problem is not going to fix it, and in fact continuing to regard these people as beneath you is only going to accelerate the downfall of this system.