Peer review:
1. dramatically reduces the pace of progress
2. exacerbates publication bias
3. creates a false sense of accomplishment
4. creates a false meritocracy
5. creates many perverse incentives
(3) and (4) unfortunately wrap into grant financing as well. The merit of your research isn't measured on its impact on the human condition, but on it's 'impact' factor (a measure derived from your publications).
The gatekeeper problem here is pernicious as well. If you become a highly cited author your ability to get/maintain financing improves. You also become a gatekeeper as a peer reviewer. Which means that you are now strongly incentivized to accept papers from people who cite your work or align themselves with you, and reject everything else.
What is absolutely amazing here, is that the peer review process is opaque. It is my belief that if you knew who reviewed which papers you would quickly discover that mild to severe abuse of peer review is the norm, not action by a handful of bad actors. This is because the entire academic reward system is wrapped into the process. Getting your name on a big paper can have lifelong ramifications on your ability to get grants, start companies, do consulting work, etc.
Peer reviewers should probably get paid for the work. If they don't get paid then their incentive to do review must come from somewhere else, vague notions of improving the field don't cut it. Peer review should obviously be transparent. Some people might be uncomfortable signing their name to a paper rejection, but its time to get over that. A small payment might help reviewers overcome this discomfort. It is bizarre that peer reviewers don't get paid. Peer review is valuable work if done right, and without payment all the reward of being a peer reviewer comes for the wrong reasons.
Finally, I agree with some other comments. Publication should not be contingent on peer review, it should come first. This would increase the pace of progress, reduce publication bias, reduce the false meritocracy, reduce the ability for bad actors to censor research, and more. The cost would be a larger number of publications, but perhaps this would help people realize that many of the publications coming out right now are of little value.
Transparency is badly needed in many facets of academic research. My company made a site that helps bring transparency into literature review (sysrev.com).