You're conflating two things.
One: there is government waste and inefficiency.
Two: this department is a reasonable solution to this problem.
Most people will agree to one, but two doesn't make a lot of sense. There is already a department that does what DOGE is tasked to do. Which means that we will have a redundant department of efficiency. And that redundant department of efficiency will have two (billionaire) heads. Why would a billionaire take a government salary (pennies to them) so that they can run a department like this? Well, a billionaire whose primary wealth is tied up in businesses that rely on government subsidies and government action to control the market in which it operates would see this as a great tool to help influence decisions that impact their business. If you can't infer, I am talking about Elon Musk.
Elon Musk has companies which are facing lawsuits for illegal union busting behavior, and not complying with appropriate overtime pay and not giving legally required breaks. His idea of efficiency is making the laborers work harder. When the entire air traffic labor force is shouting that we work TOO MUCH now, this runs absolutely counter to his ethos of more hours and more output per hour. If his thinking is applied in aviation, where we work with contingencies and checklists to ensure multiple layers of safety: we will see accidents that can be directly attributed to these policies. Not to mention a much LOWER standard of living and working for Air Traffic controllers who stay with the career field because we will again be expected to do more with less.
You can complain about staff support specialists, but do you also want to be responsible for deciphering every relevant change in the 7110 and briefing the staff in a meaningful way? Do you want to run through the overtime roster while youre working 4 sectors combined during a thunderstorm? Do you want to be responsible for ensuring that the staff is briefed on fire evacuation procedures at the appropriate intervals? And do you want the government to be able to fire you when you "underperform" any of these job duties? All of the job duties these support jobs perform will just be thrust upon the line controllers and we will not get paid more for it. They are here to cut costs period.