DoE definitely should go, and I think it would even be politically feasible if people understood that would mean more local control over schools. But most people think DoE == public schools, so even abolishing DoE scares away most people.
Yes, it has been a Republican talking point for years (much like repeal Roe v. Wade), but I think it is merely a ploy to fire up the base; they will never actually deliver on it since it is politically unpopular. I try to stop listening to what politicians say they will do and focus on what they actually do. Republicans seem to focus instead on pandering to corporate interests when they achieve power.
For sure the public doesn't understand what removing the DoE would mean. I don't know if it is because Republicans do a horrible job explaining things, or if it's the lack of media support to sell it.
You'd have to ease people into it for sure.
Start by abolishing the Department of Education. That's something Republicans have talked about many times (But do they, when they have the ability). Return all education choices to the states.
The states could then kill their Department of Education, giving power to the county level. Now citizens could be given a huge portion of their money back if a city decides to remove education from their property taxes. This would appeal to many.
Those are good goals too!