A major journal in my field is published by Elsevier. How can we move the field to a less objectionable, more open publisher?
As a publisher who has launched an OA journal and seen what it takes to do so, I see the biggest problem for starting any journal being lack of an Impact Factor (IF). I've polled authors at ECVP and about 90% said IF is what determined where they submitted. Everyone knows why this is, so why don't academics turn inward and try to undermine this reliance on IFs? Doing so would insert true competition into the journals market. Maybe academics have tried to do this, though I've never read of any real substantial attempts. I could be wrong. The second problem after IF and before funding is loyalty. Ed board members are often tied to multiple journals and in those first few years you really need those big names on your board to commission for the journal. Big names often care more about another journal or just don't care. 

You need a publisher, be it nonprofit or profit, that can put in the work to promote the journal and help commission. Financially, OA journals are very easy to start. Subscription journals require more financial backing, which perhaps could be gotten through grants if you don't want to be tied to a commercial publisher. The answer is long and requires lots of discussion and more important, commitment.