In fact, isn't that the problem, that quantity of publications (and citations) is used as a metric for continued employment? Those awarding research grants, etc., don't provide a free open-access publication as the place publication is required.
It seems governments could fix this in favour of the public quite easily; require publication in an open access journal of all research (papers and data) with any funding from the public purse.
Governments and other major research funders are indeed beginning to do what you recommend and mandate various kinds of open access terms on publications. There are multiple options, some of which are reasonably favourable to the incumbent publishers (e.g. delayed OA release with an embargo) and some of which are not (e.g. it goes straight into our public database of funded research on day one). What the end game looks like isn't exactly certain at the moment.
You usually have to apply, and spend countless hours getting permission to spend the money, fill in forms, etc. It's all just too complicated.
Academics don't have time, either to find where the OA content is, or how to access it, and nor do they have time to organise their content to end up open access. It's much better for academics if everything is in one place, and trivially accessible.
Journals should be paying sci-hub to maintain a single repository where all articles are accessible for everyone, trivially, then switch to a model where universities pay only for the services journals are actually providing (in some cases, from what I can tell, none).
The deposit problem is, as you say, a lot knottier. I personally suspect a linked network of institutional repositories with a twelve month embargo may end up as the end state of the current process, which I don't think is good enough.
Sci-hub provides a sociotechnical means of doing differently, and that itself justifies its existence, I think.
Leaving aside the societal merits, our incentives are to get published, then get read. Sci-hub helps with the latter, and doesn't harm anyone except the publishers (who most academics cordially dislike anyway).