The problem is not access, it's the confusion around what journals are for. Researchers use journals for finding information and publishing information. But research
institutions use journals as a hedge to avoid having to figure out what is "reputable research / researchers" and what is not. Those are two completely different use cases, so you need a solution to both.
The simplest solution would be to separate those solutions, so there's one system for vetting information and researchers, and one system for finding or publishing information. Those can be different businesses or community-led efforts.
There's also the aspect that publishing does not have to be inherent to access. One thing is finding who can publish; another is vetting their information; another is editing and layout; the final one is publishing. And then on top of that, is access. All of these are discrete steps which can be done independently of the other, by different organizations.
App stores are one example. There's not one company putting out a million apps. They set standards and do curation, but the apps are completely constructed by an outside party. The app store handles publishing and access, but the publishing aspect is mostly just hosting.
Another example is an Operating System. They don't restrict publishing or access, but they do provide a platform on which to do all the things needed to make or use data. Technically you only pay once for access to the platform, and other parties deal with the rest (which may involve payments).