That's a straw man argument you're setting up there as you're probably not well informed on those departments and how efficient/inefficient they are.
I can attest that the Dept of Education has done nothing but drag down our collective educational system as a country.
Ever since it's inception and well into it's programs, there has been more money spent on underperforming schools and children than ever before.The return promised has never been fulfilled. We cured polio, smallpox, and sent men to the moon without the Department of Education. We haven't done anything but drop in the educational rankings worldwide.
Talk to a teacher. See how their hands are tied due to regulation this and regulation that. Teachers that want to teach do not want the Department of Education. Teachers that have tenure and like a paycheck want it to stay.
Getting rid of those departments doesn't mean the end of it. Get real and intelligent. They're not going to just wither up and go away. The onus shifts to those who have a dog in the fight.