[Someone]

Devils's advocate: if no "private parties profiting from providing access to publicly funded research" should exist, shouldn't you be opposed to "Most research papers are findable via Google.", too? Google is making profits from publicly fundable research, too.

IMO, there is nothing wrong with companies making a profit from providing access to publicly funded research; the only problems are lack of competition and (IMO consequently) profit margins that society finds unjust.

The former, I think, is being settled. More and more new research is effectively open access. The latter, then, will follow.

And to your second point: there used to be companies that had university-like environments where short-time profits do not rule the day (AT&T, Xerox, Philips Natlab, Apple's ATG). Nowadays, Google and Microsoft Research still have similar groups.