Let me try it this way.
Here's Brad DeLong:
"So, by continuity, somewhere between policies of austerity that that produce deflationary depression due to an excess demand for safe assets and policies of fiscal license that produce inflationary boom caused by an excess supply of government debt, there must be a sweet spot: enough new issues of government debt to eliminate the excess demand for safe assets and so cure the depression, but not so much in the way of new issues of government debt to produce destructive inflation, right?"
Is continuity assured?
(I'm sorry to pick on Brad, but he made the mistake of writing too clearly, and laying out the implicit assumption too explicitly.)
Then all of a sudden there was a discontinuous change in the market for private debt. Perceived risk jumped up. Demand jumped down. Prices jumped down. And those discontinuous changes caused a financial crisis, and that financial crisis caused the recession.
It wouldn't have happened under continuity. The prices of private debt would have fallen slowly as the stock of debt increased, causing the issuance of private debt to slowly grind to a halt, with no financial crisis.
Why couldn't the same discontinuous change happen in the market for government debt?
Yes, there is one big difference. Governments that own a printing press can and very probably will act as lenders of last resort to themselves. And they might not act as lenders of last resort to the private sector. That means government debt is less vulnerable to a liquidity crisis than private debt. But there are solvency crises as well as liquidity crises.
If people suddenly want to hold less government debt and hold more government money instead, the central bank can simply swap money for bonds and fix the liquidity crisis. But if people suddenly want to hold less government debt and hold less government money, that won't work.
Why can't there be discontinuity in the market for government debt too? Why can't there be a Minsky Moment with government debt too?
Yes, the results would be different: an inflationary crisis rather than a deflationary crisis. But it's still a crisis.
You want a model? Sure. Give me your model of the discontinuity in the market for private debt, I will replace "private" with "public", and hand you back the same model.
Update: and if you say that an economy at the ZLB is qualitatively different from an economy not at the ZLB, and many people do say that, you are saying that there is a discontinuity on the boundary between the ZLB and the non-ZLB equilibrium. "Too little, still a little too little, very nearly enough, far too much!" Back to (Ryan Avents?) Ketchup Theory. Monetary policy is easily and quickly reversible in a way that fiscal policy is not. The central bank can easily and quickly buy back the stock of money it has issued; the government cannot so easily and quickly buy back the stock of debt it has issued.