What is it that gives the publishers leverage in this negotiation?
"Science advances one funeral at a time."
When prestige signalling is no longer tied to those journals, the publishers will lose their leverage.
I'd love to see a group of researchers band together and all agree to publish in some alternative to Nature. Leave no 'prestigious' authors publishing there. And let the cabal that are stuck in that thinking realize everyone is walking away from them and making them irrelevant.
It's theoretically possible that academics collectively decide that closed-access journals are off limits and start over with a new set of open access journals - but I suppose they are rather doing what they can (or relying on university administrations) to push for publishers to change this "exploitative" economic model (where the public essentially pays twice - first to do the research and then second to have access to read it).
While academics define themselves by innovation, there is a very strong adherence to establishment and institutions (like the traditional peer-reviewed publishing model, though there are good arguments for this as evidenced by problems caused by unpeer-reviewed preprints that have hastily made their way into public discourse during coronavirus). The hope is that academics don't have to throw away a method for signaling publication quality (or rather, perceived potential for impact) that has pretty been a part of the entire history of modern science. Whether it's a true mark of quality or not is a different matter; the most prestigious journals also have the highest retraction rates.
In Europe, funding agencies have begun requiring publication in open access journals as a stipulation for receiving funding, and this has driven journal publishers to provide an option for authors to pay for the publication (which is paid out of grant funding). However, this is not still not pervasive because there are still many funding sources that don't require it. Publication fees for authors are quite steep (around $3k per article; a productive lab may publish something on the order of 10 per year), and if the grant doesn't specify that they would cover open access publication fees then researchers tend to squeeze everything toward covering costs to do the research (equipment, personnel, travel to meet up with collaborators or other researchers).
I once spoke to someone at a conference on the subject of publishing (maybe some 6 years ago) who told me about a journal coup where a large majority of the editors and staff for a journal quit and started their own journal which was well received in the field and quickly becoming the journal of record. I can't quite remember what field it was, but I think it may have been related to neurology (non-human IIRC, as the individual had just given a presentation on cuttlefish cognition).
Anyway, for a close community of researchers on a specific topic, it's possible that they can make such agreement (to start over with another journal) - but in general it would be quite difficult to coordinate such a consensus in a larger community.
Even if journals are not considered prestigious in the traditional sense (often measured by what is called an impact factor - related to the average number of citations its papers receive), publishing in them still marks membership in a particular community (e.g., certain types of papers or papers addressing a certain topic have historically been published in X; following the succession of your academic forebearers).
So there is a lot of incentive not to scrap the current journals and start anew.