After the legislators discovered what they had voted for, many said they were horrified.

Tucked away in the new pages was a provision that would have made it much harder for local bodies like school boards to raise money through property tax increases.

“Michigan schools will likely suffer the brunt of the impact because the vast majority rely on periodic voter approval of local operating levy renewals for property taxes,” the ratings agency Moody’s wrote of the measure the following month.

“I was fooled into voting for something I opposed,” said Dave Pagel, a Republican representative. “I consider it the worst vote I’ve made.”

The chief culprits, according to Mr. Pagel and others at the state Capitol when the bill passed, were lobbyists closely tied to the DeVoses.

Tony Daunt, a spokesman for the Michigan Freedom Fund, a nonprofit headed by the DeVoses’ longtime political aide, and whose political spending arm they have funded generously, said the group was “part of the discussion process with people in the legislature” about the proposal and “had consistently expressed support for the policy.”

The law was later blocked by a federal judge, but the group has vowed to try again.

Radical Suspicions

Ms. DeVos’s advocates see in these fights the toughness to take on entrenched opponents of expanding reforms like charter schools and vouchers.

In promoting Ms. DeVos in The Washington Post, Mitt Romney, the Republican Party’s 2012 presidential nominee, emphasized that her wealth gave her the independence to be “someone who isn’t financially biased shaping education.” He added, “DeVos doesn’t need the job now, nor will she be looking for an education job later.”

But critics see someone with an unmistakable agenda. “The signs are there that she will do something radical,” said Jack Jennings, a former general counsel for the House education committee. “Trump wouldn’t have appointed this woman for this position if he didn’t intend something radical.”