In academia, promotion, tenure, funding, and pretty much everything someone needs to keep their job is tied to publishing in peer-reviewed journals. If I self publish I won't keep my job. If a university ran a website for publishing, they would have to reimplement the peer review process, and often there may be ~a dozen people worldwide qualified to review a particular paper so it's not just that others from within the university could review work. If a university is implementing all that, they have basically become a publisher and likely have costs they'll want to try and recoup and could foreseeably implement a fee to publish.

Don't get me wrong - the journals have a predatory and exclusive model that should be dismantled. But until we fix promotion, tenure, and funding pathways in academia that have enabled the publishers to become what they are there will always be these problems -- any other system that pops up will not get widely used because academics will be disincentivized from using it (as discussed in some other comments here).