If each academic research institution can run its own website, there is no particular reason they could not publish every last research report they produce themselves, and use a common, open-source program to handle the draft commentary, editing, and peer review.
A government publication house would overly centralize things, and would organize research by geography and politics rather than subject matter. Science is a worldwide endeavor, and ought to be insulated from political influence as much as we are able to do so.
And we certainly don't need any more juicy nexus points, dripping with extractable rents, throbbing and pulsating for the benefit of the leeches.
One academic publications office per university or independent lab would certainly cost a bit more in overhead, but nowhere near as much as what is currently lost to the publisher conglomerates for both publication and access.
The aggregation function of the journals could be fulfilled by a nomination and community moderation process among the researchers and students of any particular discipline.
Publishing via university web site would also allow linking to related files, such as the raw data, conference video, audio, or transcripts, executable simulator or demonstration programs, and related research, all with permanent, canonical URLs.
While I am not an academic, I notice that many of the journal-curated publications (that I can read for free) have only static diagrams, whereas if you go to the researcher's web site, you can see animated illustrations, executable programs, and lecture slide decks or videos. When the mission of science is to not only advance the frontiers of human knowledge, but also explain what you did and to disseminate the new knowledge to others, a plain old PDF locked up behind a paywall might not be the best way to do that.