I agree, the process of peer review is something that should probably be done by academic institutions with neither corporate nor governmental oversight, and seems like something they're mostly happy to do, as evidenced by so many journals getting the work of expert scholars as peer-reviewers for free. As someone in a different comment thread pointed out, part of the barrier for such a move is that funding decisions build journals in as a gatekeeper, so effectively funding is locked behind publication. It's possible that simply removing the publishers from the equation would create the result everyone but them seems to want "automagically"