No, universities can't do this. They can't compete with publishers directly because publishers can decide to cut off the supply of journals to libraries.
They also can't set up a competing independent paper service because there's no way New Journal X can compete with the brand recognition of Nature or Phys Rev D.
The publishers have a de facto monopoly on the prominent brands. That's why it's extortion, and not a service. The only service provided by the publishers is access to the goodwill associated with the brand.
What universities can do - and are starting to do - is to set up alternative publishing systems that will start to bypass those brands. Arxiv is the most famous examples, but increasingly communities of academics - not universities - are creating their own online enclaves, with the optional prospect of live debate about papers instead of the current somewhat dysfunctional formal peer review system.
Eventually the goodwill for many disciplines will move to those online enclaves, and that's when publishers will lose their leverage.