Bleg: What is the issue endogenous money people are after?

I have a question where I would love some help 🙂

Via Marginal Revolution I noticed that there seems to be quite a war between economists regarding the financial sector.

However, I actually don’t see where there is an issue in the argument that is going on.  The way I see it Nick Rowe has it right with his comments here and here, and in turn this illustrates that even though a central bank doesn’t explicitly constrain the “stock” or the “supply” of money, their cash rate (and its relation to the natural rate) and inflation target (given an output gap view of the evolution of inflation) provide an implicit constraint.

The way I see it, mainstream theory and “endogenous money” theory people can both assume that the quantity and/or supply of money is endogenous, and that investment=savings-current account.  So where is the difference, what does it tell us, and how is it testable?