Well, a lot of the financial plumbing breaks because it is based on Treasuries being a risk-free, AAA rated assets. Treasuries get downgraded and prices collapse as institutions are forced to sell because of ratings issues. Other credit markets are dragged into the same vortex and credit availability gets pulled way back. Housing prices decline. The tides go out on an overleveraged financial sector and an overheated financial market that is priced for perfection, major stock pullback perhaps as high as 50%. Consumer confidence collapses and businesses pull back amid the uncertainty sending us into recession Inflation remains stubbornly high crimping real spending power. The usual fiscal policy boost doesnt materialize in the short-term as Congress is handcuffed from spending, both by partisan politics and a sharp rise in the cost of debt (spending probably actually decreases in the short-term). We enter recession, not as deep as the financial crisis or COVID, but probably prolonged maybe like the extended malaise of the 1970s. Unemployment probably goes to 9-10% and hangs out there for the rest of the Biden administration.
Empirically, just the threat of of not raising the debt ceiling nearly caused a recession in 2011/2012 and things were a lot less overheated and we werent in the middle of a global pandemic.
EDIT: The thing that I think is hard to appreciate is that even a short default means the treasuries are going to be rated a default. That means massive forced selling in massive size, there isnt a choice of “oh we’ll just wait it out and see what happens”. The documents and mandates of many of the large treasury holders and/or bank regulations will force massive and immediate selling with limited buyers and will force massive selloffs in other risk assets (if treasuries now yield 8%, how can a US based company debt yield 3%? It cant, price has to go down). There are likely very bad unintended consequences that will have major economic ramifications with limited ability to use the tools that we usually use to stave off recessions.