I'm an editor of a journal that has one of (the?) highest IFs in its field, among other things. Lots of experience with scientific publishing.
I've had lengthy discussions about this with my colleagues, and I don't think the problem is with peer review per se, it's with how peer review, and by extension, scientific literature, is interpreted and utilized. It's not just with the lay community, it's among scientists themselves, as well as scientific consumers as in engineering, healthcare, etc.
People now treat science, and scientific publications, as if it's a farm or factory. The produced commodity is the peer-reviewed paper. Once research, via a paper, is peer-reviewed, it's blessed as the truth, and if it's not, it's worthless. The goal of many institutions is now not to produce sound research per se, but to receive money (largely due to indirect cost income) to produce the commodity of peer-reviewed paper. Once a paper is accepted as in press, it can be cited as if it were truth, which provides a building block for something else. The truth per se doesn't matter, only the citability of it as peer-reviewed.
I realize this sounds a bit conspiratorial but I do think this is basically how it works at this point in time, at least in certain fields. Peer-review is overvalued, as are single papers, and contributions of particular researchers in many cases. That doesn't mean they don't have any value, just that their real value is probably much less than we assume.
I agree that peer review can be improved a lot, and will likely always have an important role in some form, but I don't think fundamental problems in the field will go away unless there's a change in perspective on the real scientific process as a whole, to something more nuanced and gradual.