What is the serious downside of open internet centric review?
You could of course make it double blind, but that seems hard to enforce in practice in such an open setup, and still, hyped papers in fashionable topics would get many reviews while papers that are hardcore theoretical, in an underdog domain, etc. would get zero.
Finally, it also becomes much more difficult to handle conflicts of interest, and the system is highly vulnerable to reviewer collusion.
This is less of an issue with systems where there is little monetary value attached (I don't know anyone whose mortgage is paid for by their Stack Overflow reputation). Now imagine that the future prospects of a national lab with multi-million yearly budget are tied to a system that can be (relatively easily) gamed with a Chinese or Russian bot farm for a few thousand dollars.
There are already players that are trying hard to game the current system, and it sometimes sort of works, but not quite, exactly because of how hard it is to get into the "high reputation" club (on the other hand, once you're in, you can often publish a lot of lower quality stuff just because of your reputation, so I'm not saying this is a perfect system either).
In other words, I don't think anyone reasonable is seriously against making peer review more transparent, but for better or worse, the current system (with all of its other downsides) is relatively robust to outside interference.
So, unless we (a) make "being a scientist" much more financially accessible, or (b), untangle funding from this new "open" measure of "scientific achievement", the open system would probably not be very impactful. Of course, (a) is unlikely, at least in most high-impact fields; CS was an outlier for a long time, not so much today. And (b) would mean that funding agencies would still need something else to judge your research, which would most likely still be some closed, reputation-based system.
Edit TL;DR: Describe how the open science peer-review system should be used to distribute funding among researchers while begin reasonably robust to coordinated attacks. Then we can talk :)
i.e. trolls, brigades, spammers, bots, and all manner of uninformed voices.
(Actually, we already have the "open publishing" you are suggesting - it's called Blogging or social media.)
In other words, if we have open publishing, then someone like me (with zero understanding of a topic) can publish a very authentic-looking pile of nonsense with exactly the same weight as someone who, you know, has actually done some science and knows what they're talking about.
The common "solution" to this is voting - like with StackOverflow answers. But that is clearly trivial to game and would quickly become meaningless.
So human review it is - combined with the reputation that a journal brings. The author gains reputation because some reviewers (with reputation) reviewed the paper, and the journal (with reputation) accepted it.
Yes, this system is cumbersome, prone to failure, and subject to outside influences. It's not perfect. Just the best we have right now.
That's fine. I don't read eg Astral Codex Ten because I think the reputation of Substack is great. The blog can stand entirely on its own reputation (and the reputation of its author), no need for the publisher to rent out their reputation.
See also Gwern.net for a similar example.
No need for any voting.