My understanding is that the universities have contracts with the journals that mandate the professors and researchers publish in the journal with almost no exceptions. You’re putting the blame in the wrong place.
Universities rarely tell scientists where to publish. That is determined by the scientists, the quality of the paper and the editor of the journal.
Scientists and PhD students need to publish in high impact factor journals for their work to be recognized, for promotions, graduation, etc. There is a lot of work that goes into a scientific publication. It's not a blog (which most people equate it to when they say why isn't it free).
Scientists can make their publications available on their personal website. Generally google scholar will give you a pdf if its available. Some labs maintain papers on their site, other scientists don't. Generally, finding older publications is a challenge.
I've commented on this earlier. Asking the researcher to not publish in these journals is pointless. You need to legislate access. But in general, most scientists will have access to these journals from their university libraries (at least in US/Canada/Europe).
I've never encountered a mandate to publish in certain journals because the university has contracts with them. On occasion there are incentives like breaks on open access fees essentially because the university pre-paid, and there are field specific norms on where to publish.
This statement makes me think you significantly misunderstand how scientists / universities function. The scientists (graduate students, postdocs, professors, staff) are all salaried employees of the university. The university does take some of the grant money given to lead professors (e.g. to pay the aforementioned salaries).
If someone were to snap their fingers tomorrow and make it so that scientists could publish their work without having to deal with these for-profit publishers, universities would save millions of dollars (per school), taxpayers would save money, and the general public would also have greater access to the work that they themselves funded through public grants.
[1] https://www.universityofcalifornia.edu/news/why-uc-split-pub...
There was also a big fight over prices with Nature about 10 years ago.