If Github and Gitlab (and maybe Bitbucket) had the balls to build something like that, I think things would improve. Users would be impacted, so it would be essential to get some help on the PR side from established projects and bodies like FSF, Python Foundation, and so on. Getting the big companies onboard should be easy enough - just give them a flat-fee yearly subscription option that they can spin in a nice Bloomberg release on how they support innovation and blablabla.
You would likely still see cheapskates setting up their internal mirrors and so on, but with the right price structure, I bet most medium-to-big shops would accept it and start generating substantial amounts for OSS projects.
One problem I see is that most open source doesn't get consumed as source code but via language package servers (Rubygems/NPM) or OS distributions (Debian packaging Redis).
For other packagers and OSes: packages don't happen out of thin air: someone has to write recipes and scripts, and test them. So that's already some revenue right there, especially from the big players. Companies like RedHat and Ubuntu could just cut a flat $10m check every year, distributed to projects proportionally in accordance with OS-provided stats like popularity-contest. That's the easy option. Ideally, you would also have some buy-in from the major OS distributions that could somehow "trickle down" the model to their own packages (say, linking a VCS repo to an OS package and reporting activity accordingly: "You asked to apt-get install Redis, but I've checked with a webservice and you have already downloaded it 2 times from Github.com today: time to pony up!". In most cases, this sort of link is already documented formally, somewhere in the package definition.)
Like newspaper paywalls, it doesn't need to be perfect; it just has to be enough hassle that most people who can afford to make the pain go away, will just do that.