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It puts them in the tough spot of declaring Doge essential. That gives a huge opening for the Democrats to attack, when 80% of the government is shut down, that there is meaningful fraud in what Trump kept open. It’s also going to be hard to get someone to come in to unlock things when they aren’t getting paid. Even essential employees don’t get paid in a shutdown, they get back pay after like everyone else when Congress approves it.
There isn’t management for these jobs though. The people are the management. Both sides of the table are the same in the public sector.