Exceeding the debt limit does not mean default. It means that all further spending immediately gets reduced to the level of tax revenue; implying a 30% drop in spending.
It is up to the treasury to prioritize where the cash on hand goes.
They say they have neither the technical ability (the software can't handle the use case) nor the legal authority to prioritize payments (a de facto line item veto). They might just be lying to us, but if you believe they are capable of that, you also believe they totally would default just to prove a point.
Besides, if the government is going to pay you in a couple weeks, we swear, that would be a debt. It has just been shifted away from the people who want to lend us money voluntarily.