One way to cut out the middle man would be to convince journal editors to run sibling journals alongside the existing journals, so for each "Transactions on XYZ" there would be an "Open Transactions on XYZ" (as close in title as is legal). Importantly, each sibling journal would be run by the exact same academics (who are doing the real work on tax money anyway), and according to the same process as the original journal, just without involvement from a traditional publisher. PDFs would be hosted on a site like arXiv. The goal would be that submitting to the open "sibling" would be the obvious rational choice (same people, same decision-making, no fees, open access), which in time even the funding agencies and tenure committees would have to acknowledge.
So rather than a reviewer lending their reputation to a journal, and that journal then conferring a stamp of trustworthiness onto an academic work, reviewers lend their reputation directly to the works they review. The works themselves can still be shared far and wide, e.g. via ArXiv.
Of course, inertia is still a massive force to work against.
The biggest problem I see with the proposed system is that it's unfortunately often easy to recognize who wrote a paper (which is bad for ordinary peer review already) and in a personal endorsement system this would lead to collusion among researchers with low integrity. You wink through my papers, I wink through your papers. Also, imagine you don't endorse the paper of the senior researcher in charge of your postdoc funding...
(Something similar goes for researchers in charge of your postdoc funding: how many co-authorships are earned, and how many are ways to game the current system?)
I'm certainly not saying that a public endorsement system is the end-all-be-all and won't have its own problems. However, I do get frustrated every now and then by the institutional inertia that arises from holding new initiatives to higher standards than existing ones (see also: using the Impact Factor to evaluate academics).
There are many things that needs to be addressed to improve science, not just peer review. Hiring policies need to change and indicator counts need to be used more adequately, empirical studies need to be pre-registered and the p-value needs to be 1% or lower. Peer review is just one factor and mostly a monetary issue. If all research institutions would spend enough money to make all commercial journals available for everyone, there would be no particular problem with peer review. The problem of many researchers in poorer countries is that they don't. The problem is not that peer review does not work.