I would say Shmidt's agenda is more neo-liberal than progressive. Not that I endorse it, but corporations pushing an agenda is not new[1] and not limited to the left[2][3], so not even conservative media will stand in their way either since that has been happening for a long time on the right.
Hobby Lobby and Chick-fil-A aren't out sending hundreds of millions financing think tanks and lobbyists in Washington, they are just trying to run their own business with their own agenda. And the scale of the companies are orders of magnitude smaller than Google, both are medium sized family owned businesses.