The economics of the debt ceiling are complicated, but the bottom line is that the ceiling is far more forgiving than is commonly understood. When the ceiling is reached, little happens except that spending slows down. Revenues continue to flow into the federal Treasury and are massively more than needed to pay interest on bonds and principal coming due (which then, of course, enables reissuance of new bonds under the ceiling). So, there need not be any default.
But like a person on a credit card spending spree who reaches their limit, the U.S. government has to establish spending priorities. Unthinkable, of course, would be to default, so it’s other spending that has to conform. So-called entitlements such as Social Security and Medicare are probably at the head of the line, followed by discretionary spending such as providing for defense, running the government and doling out grants. Spending in this category is easier to limit. Such programs typically require authorizing legislation, appropriations and available cash. The first establishes the program in theory, the second provides money for the program, again in theory, and the third — well, you get the picture.