Pilot here. I fly little airplanes for fun.
Much of the ATC system is already understaffed by a good amount. Becoming a controller is not super easy, nor is it for everyone. You have to pass a fairly stringent medical test, background check etc, and then you go to their academy for a while. After that, they will assign you wherever they need you, which may be nowhere near where you actually wanted to live. What you make depends on where you are and what you do. So for example, if you man a small airport in the middle of Idaho, you make less than if you are an approach controller near JFK. Each position that you might work requires its own training and certification, and I don't just mean each physical location I mean like each chair, each working position. That's because each sector of the airspace has its own quirks, where traffic usually comes from and usually goes to, defined airways and GPS points and procedures and that sort of thing.
My point with all this, is that hiring and training new controllers is significantly harder than most other jobs. To use the earlier example, and approach controller near JFK is doing one of the most difficult and stressful jobs in aviation, and is easily making six figures. But to get that guy there, getting him trained and certified on everything took years and hundreds of thousands of dollars. If he resigns, whatever salary you save on him is a drop in the bucket compared to what it will cost to replace him.
Private sector controllers are a thing. Many small airports contract out their tower operations, but this is generally done at small regional airports.
Don't get me wrong, there are plenty of places where fat needs to be trimmed in the federal government. I don't believe ATC is one of them. And is certainly not a place where I want fat aggressively trimmed, because when you lose experience you reduce safety.
