Let's remember what we've got here. Productivity is output per labor hour.
On the left, you really don't want to argue that people should be earning their output per labor hour. Just stop and think of what that entails for other factors of production.
So why cite productivity? Because so far we have not come up with a good way to calculate the marginal revenue product of labor and publish regular statistics on it. We certainly don't have a good way to do that for the marginal revenue product of the group of workers earning around the minimum wage. But these metrics ought to grow together, right? Marginal revenue product of labor and productivity ought to move around in the same ways.
So I maintain (and Elizabeth Warren appears to maintain), that it's a good metric to keep track of. And indeed, productivity growth has been closely related to wage growth (until recently, where there have been some interesting divergences). I see the EPI graph looking at the growth of the minimum wage if it grew with productivity and think "Gee - that's really interesting. We always talk about the real minimum wage, but that's actually incomplete - productivity growth is important for thinking about whether a minimum wage is binding at a given point in time. A steady real minimum wage ought to be progressively less binding over time."
I don't agree with Warren on everything, but she's a smart cookie.
I don't think she wants a $22 minimum wage. Exhibit A in evidence is that (as far as I'm aware - correct me if I'm wrong) - she's never said she wants a $22 minimum wage. That's an awkward little factoid that no salacious post title at Huffington Post or Daily Kos is going to undo (again - correct me if I'm wrong on this).
She's a very intelligent person. I imagine she sees the EPI/productivity graph in much the same way I do - as indicative that workers have been losing some ground and that a given real minimum wage is less binding now than it was several decades ago when worker productivity was lower.