“I could never have imagined so much attention,” Naftuli Moster, a yeshiva graduate who now leads a group advocating for their reform, said. “We have groups that have never been interested in looking into an issue mostly relevant to nonpublic schools.

“Suddenly, when this holds up a $168 billion budget, they’re taking a closer look,” he said.

The lack of attention is one of Mr. Moster’s biggest complaints. In 2015, under Mayor Bill de Blasio, the Department of Education agreed to investigate the quality of schooling at yeshivas, at the request of Mr. Moster’s group, Young Advocates for Fair Education. Mr. Moster said that when he graduated from a yeshiva, he had never heard of basic scientific concepts, including the word “molecule.”

Nearly three years later, the results of that investigation have yet to materialize. Toya Holness, a spokeswoman for the education department, said the probe was “ongoing” and that there was no timeline for completion. Last September, the department said it had visited six yeshivas; on Monday, Ms. Holness said it had visited 15. About three dozen schools were supposed to be investigated.

Mr. Moster’s group has accused the city of dragging its feet for political reasons, out of fear of angering the ultra-Orthodox community, which is known to vote as a bloc.

The state education department’s new oversight may spur quicker action, as the department has been moving to sharpen the guidelines for evaluating equivalency at private schools. But Emily DeSantis, spokeswoman for the department, said the department was still reviewing the law and emphasized that local superintendents would still be responsible for the quality of private schools in their districts.

And state officials are no more immune to concerns about alienating Mr. Felder and his constituents than their counterparts in the city.