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.
For example, the status quo can be justified by "it seems obvious that" rather than being evidence-based. In practice, single or double blind is very often not actually blind [1], in which case it's hard to argue that it would be any different than transparency. Likewise, a new solution "creates all kinds of conflicts of interests and biases", without even considering whether the existing CoIs and biases are any better. Likewise, if people who game the system today have a low reputation, why would the same not hold true in a new system? Likewise, does anonymous peer review actually improve the quality of the reviews (e.g. [2])?
It's good not to blindly embrace something new, but I think it's important to withold judgement too, and see how it plays out in practice, and to make an effort to compensate for the prejudice people typically have towards the status quo.
And yes, absolutely agreed that much more needs to be addressed to improve science. But I also very much take issue with the idea that peer review is fine as-is.
[1] https://absolutelymaybe.plos.org/2017/10/31/the-fractured-lo...
[2] Heading "quality of feedback": https://plos.org/resource/open-peer-review/
Maybe you should consider that you're the stubborn one who links to blog posts from open access journals with a strong bias to reform the peer review system as evidence. Plos journals have this agenda because they don't have good enough pools of reviewers, which is the problem of all new journals. However, it's not as if scientists all over the world and funding authorities haven't thought about the topic. People should not withold judgement about these issues, that's an odd request and would weaken the position of scientific staff towards political decision makers who certainly won't withold judgement.
To give you an example of what is being done by funding authorities, at our university we are required by law to publish every article in openly accessible format. Otherwise they do not count at all towards our salaries, they will be ignored completely. There are public open repositories for that. The only thing that bothers everyone is that big publishers like Elsevier only allow draft versions in those repositories and there is sometimes a 1 year mandatory delay until they can be put online. That needs to change and EU authorities are working on it.