Why do big funders not act as publishers?
As you note, in a major sense this is actually already done on a large scale by the NIH in the US.  The NIH requires that all publications resulting from its funding be made freely available on PubMed under an open access license within 12 months.  While PubMed Central does not do peer-review, it does provide the other half of the functions of an archival publication, in providing hosting and curation.

By doing this, the NIH is accomplishing most of the goals you mention, because there is now much less pressure to subscribe to lots of high-cost journals.  At the same time, however, this is better for the NIH than running journals itself for at least the following reasons:


Running journals does have a significant cost, even if that cost is often hidden in free society journals and overcompensated for in for-profit journals.  Forcing compliance as a byproduct of funding is probably much cheaper.
If the NIH started running journals, it would be competing with traditional journals, and thus would obtain much less coverage than it does by just forcing public deposit of their articles.
Scientific publication is highly international, and even the biggest countries are a minority of the researchers in any significant field. There may thus be significant perceived or real conflicts of interest if a major government funder is running the peer review process for other nations.


In short, it appears to be currently much more efficient to accomplish such goals by using the compulsory power of contract clauses than by investing in competition with the existing publication ecosystem.