SOUTH ORANGE, N.J. — Kate Walker, a New Jersey mother of three, says she feels like one of the lucky ones: She enrolled her children in a Catholic school for September before a wait-list started.

But she has held off making a final tuition deposit, hoping that her town’s public school district, South Orange-Maplewood, brokers a lasting solution with the teachers’ union, which had resisted a planned phased-in return to school, citing coronavirus safety concerns.

“I’ve lost a lot of faith in the district,” said Ms. Walker, who participated in a recent sit-in outside her 7-year-old son’s elementary school. “We’ve been stopped and started a dozen times.”

Most districts in New Jersey have partially reopened, but one in four children still live in a district where public schools are closed. No state in the Northeast had more districts relying on all-virtual teaching in early March than New Jersey, according to Return To Learn, a database created by a conservative think tank, American Enterprise Institute, and Davidson College. Nationwide, only seven states had a greater proportion of all-remote instruction.

As the distribution of Covid-19 vaccines has accelerated and President Biden has signaled a push for broader reopenings, frustration among parents has grown, particularly in New Jersey’s affluent suburbs, where schools with stellar reputations are a key reason families are willing to pay some of the nation’s highest taxes.