One important thing to bear in mind is that a policy like this is not a silver bullet. Simply solving the copyright question won't actually mean people then go on to make material available.
The proposed approach would allow authors to distribute the "original" (ie preprint, postprint, author's version) - text, formatted as they wish, without page numbers, journal copyediting, etc. Those pretty shiny journal PDFs would still be out of reach.
This is actually more or less the situation that exists in the UK right now - there is the important distinction that an embargo period is involved here, but otherwise, something like 95% of academic articles can be posted and redistributed by their authors in this way. To do this, we've used funder mandates (and pushing the publishers into agreeing embargos) rather than hacking at copyright transfers on an institutional level, and it's generally worked okay.
The problem is, the articles don't get made available. Or, at least, a lot of them don't. Many institutions are finding it difficult - even with someone paid to chase academics and hold their hand through the process - to get more than about 50% of eligible papers made available through a repository. Indeed, for some institutions, even identifying those papers to begin with is challenging.
In the event that every university in the US overnight made every researcher sign a pre-emptive copyright license like this, I suspect you'd have the same problem. Yes, it could technically happen. But inertia would limit what actually does happen.