Some have, but even that depends on licensing options. Many universities have institutional repositories with folks working hard to get affiliate papers uploaded for open access, but they still have to follow the publisher's license. Some publishers allow OA upload in an IR after an embargo period. Some do only if you pay for general OA publication (extra cost on top of the regular publishing costs, although subscribe to open or read & publish deals sometimes take care of the fees). Some allow it as a matter of course. Some allow it if the author requests it at some point. Some just don't care and never allow it.
There are also university presses or nonprofit publishers, but their models often aren't that different. It should be treated as a public service, especially for research given public monies to be completed, but it's currently just business as usual.
There are a bunch of field-specific preprint services like arXiv: PsyArXiv, SocARXIV, engrXiV, AgriXiv, etc. The OSF also hosts preprints for various disciplines.
The important thing to remember is that preprints are not peer-reviewed and have not been vetted in any way. A paper may change a lot (or just a little) between preprint submission and final publication. A recent paper of mine had a few sections added for clarity, which wouldn't appear in a preprint.