That is correct. I have done this for every publication I've ever put out (O(10)). There may be exceptions but this is certainly the dominant paradigm in the physical sciences.
My two cents: this is bigger than any University. It will require a government funding agency (likely, NIH or NSF) to take a stand vis a vi the public availability of research produced by public funding.
The only problem is that the method of open access publishing is usually to pay a few grand to the publisher for the privilege.
And that is money that should be going to research.
Any papers you plan to use in the Research Excellence Framework for 2020 (for those not acquainted, an exercise every few years where UK universities get to say "look how good we are, we published this many papers, please keep funding us") must be available as open access. It isn't necessary for these papers to have gone the gold open access route (where you pay a *!&^%-ton to the publisher), so you could instead upload the non-published version to a preprint server (which is a lot cheaper).
conversely if a for profit could effectively monetize a 100% open access journal, it would be totally fine.
Additionally, many copyright transfer agreements expressly allow an institutional archival system.
Everyone in the physical sciences uses arXiv.