0. A reduction in X spending by the government means a reduction in X income to businesses and households.
1. If you reduce the federal workforce, the private sector has to absorb them.
2. For the private sector to absorb them, they need the money to pay them.
3. If you say, cut $500 billion in salary from federal government, the private sector needs to come up with $500 billion extra income to pay those people.
4. The private sector will lose $500 billion in income because generally people spend what they earn. All those federal employees were buying from the private sector.
5. The private sector usually pays better than the government, therefore for every dollar "saved" in the government you probably need $1.5 dollars in revenue to absorb them in the private sector.
So given the facts above, I want to ask people who celebrate this, where will the private sector get the money to absorb these people when they lost in aggregate the same amount in revenue? The answer is they need to borrow. Which means the private sector racks up more debt. The debt they rack up will be exactly the amount (or more!) that the federal government doesn't spend.