On the campaign trail, Donald J. Trump depicted the nation’s public schools as purveyors of an extreme ideology on gender and race. One of his proposed remedies has been to revive a Reagan-era call to shut down the federal Department of Education, founded in 1979. “We will move everything back to the states, where it belongs,” he said in one speech. “They can individualize education and do it with the love for their children.” Democrats have vowed to resist the effort. But is...