That should be only sort of true.

Some of the very high-profile journals are run by non-profits, including: Science (American Association for the Advancement of Science), PNAS, (National Academy of Sciences), eLife (HHMI/Max Planck/Wellcome Trust). A slew of more specialized journals are run by societies too.

In theory, they should be willing to lead the charge. In practice, I think they are largely dependent on income from the journals for a lot of their operations and so are reluctant to rock the boat.