Grant giving institutions could adopt something like NIH's NIH Public Access Policy[1], which requires articles to be available to the public within 12 months of publication. It isn't perfect, but it is a step in the right direction. It essentially causes everyone in that field at once to switch over to the new system, so that research don't need to stick their career on the line to make a statement about publishing.

* https://en.wikipedia.org/wiki/NIH_Public_Access_Policy