Some, but not all, universities do have institutional repositories for their own work. But they are at best inconsistently used. Some academics also (when allowed by the licensing agreement of the journal) put their own research on a personal/lab website. But again, this is inconsistent, and really relies on secondary solutions like Google Scholar to be usable (unless you already know exactly which paper you're looking for and navigate directly to the author's website).
The bottom line is that in most cases, academics rely on journals even for access to research from other folks in their own university. Things get passed around informally. And there are changing trends, such as preprint archives (like ArXiv), in some fields. But the majority of research is still paywalled.
Edit: To return to your point, yes, I was sort of discussing some Platonic ideal of "The Taxpayer". Obviously, if you live in the US, your taxes don't pay for research in Europe (only for universities' access to that research). And we could be talking about federal taxes for federal grants, but state taxes for state-run schools. But in general, the point is that the journals make exorbitant profits from labour paid with public funds.
The bottom line is that in most cases, academics rely on journals even for access to research from other folks in their own university. Things get passed around informally. And there are changing trends, such as preprint archives (like ArXiv), in some fields. But the majority of research is still paywalled.
Edit: To return to your point, yes, I was sort of discussing some Platonic ideal of "The Taxpayer". Obviously, if you live in the US, your taxes don't pay for research in Europe (only for universities' access to that research). And we could be talking about federal taxes for federal grants, but state taxes for state-run schools. But in general, the point is that the journals make exorbitant profits from labour paid with public funds.