Higher rates don’t change the amount of Treasury debt.
It does make interest payments on newly issued bonds higher, whether those bonds finance an increase in the debt limit or are rolling over maturing bonds.
The big problem is the inability to reign in spending by Congress.
Nixon was the last President with the ability to refuse to spend money that Congress had authorized. The Watergate Congress passed a law stripping Presidents of that power.
And at the same time Nixon ended the last vestige of the gold standard, which while not an effective brake at least could serve as a warning to Congress that their unbalanced budget was devaluing the dollar.
It does make interest payments on newly issued bonds higher, whether those bonds finance an increase in the debt limit or are rolling over maturing bonds.
There's a problem with your logic in a cash flow dictated by entitlement spending.