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.