These guys' models can have multiple solutions. To me, that would indicate a problem with the model. But they are happy enough just to select the most optimistic solution.
Having multiple solutions is somewhat equivalent to having different states of matter in physics. And economies definitely have very different modes, or states, that we call recessions or booms. Being able to have these sorts of local optima would seem to be necessary foe any model of an economy.
But I may not be understanding you, because we are using the same words in different ways...
As I recall they began by adopting a structural model for debt valuation that assumes no dependence on any external assets other than the debtor's. Then they reintroduce full n-dimensional dependence through a different mechanism they made up to try and force the 1-dimensional solutions into a state of consistency. It's a bit of a mess.
Actually you're right. The presence of multiple solutions is not in itself enough to make me hate the model. The analogy relating different economic regimes to different phases of matter is potentially a good one.
Not disagreeing with you, just wondering out loud if it could be revisited with other methods.