What you are missing is that my spending is your income, and your spending is mine. As a thought experiment, if everyone were to stop spending entirely and pay down debt at the same time, no one would have any income.

In a more realistic situation - when a big chunk of the GDP is used to pay down debt, the resulting decrease in spending causes the economy to run below capacity and unemployment to be high. This is why low interest or stimulus is helpful - it encourages the economy to produce at its full capacity even though spending by individuals is reduced. This is necessary UNTIL private debt has been payed down and the economy is able to run at full capacity on its own.