Scott Morrison has used the threat of withdrawing independent and Catholic schools’ recurrent funding to enlist their support in keeping non-government schools open during the Covid-19 crisis.

After Morrison’s discussions with both sectors on Wednesday, the Association of Independent Schools New South Wales has written to its members backing his position to keep schools open and noting the prime minister had warned “there were certain expectations attached to the recurrent funding provided by the Australian Government to Catholic and independent schools”.

It comes as the shadow education minister, Tanya Plibersek, has written to her counterpart, Dan Tehan, suggesting that while the opposition has been “responsible and constructive” in its comments on school closures it wants assurances the government is “properly preparing for … when they do occur” including ensuring essential services workers can continue to go to work.

On Wednesday Scott Morrison delivered a stern warning that keeping schools open is in the national interest, prompting state governments, independent school bodies and the National Catholic Education Commission to restate their support for keeping them open, even as parents vote with their feet, withdrawing children and forcing some independent schools to close.

While announcing new measures to ban indoor gatherings of 100 people or more, Morrison warned the disruption from the mass closure of schools “would be severe”, triggering tens of thousands of job losses and reducing the availability of the health workforce by up to 30%. Morrison said he would hold further discussions with the independent and Catholic sectors.

After those discussions, the Association of Independent Schools NSW chief executive Geoff Newcombe wrote to schools reiterating that it “supports his position and that of the chief medical officer, that schools remain open for the present”.

Newcombe noted that attendance is down by as much as 30% in some schools and that Morrison had acknowledged “our schools are independent”.

But he said the prime minister had “insisted that all non-government schools keep campuses open to accommodate those students who wish to engage in at-school learning”.

On Wednesday the National Catholic Education Commission also strongly backed Morrison’s position that schools should remain open, shutting down suggestions from parts of the NSW Catholic system that closures should be introduced as a matter of urgency.

The NCEC executive director, Jacinta Collins, said: “While there is significant concern and debate regarding school closures in Australia to minimise the spread of Covid-19 … it is our view that the government and public health authorities are in the best position to determine whether schools should remain open and we will adhere to this advice.

“I appreciate that, despite this advice some parents have chosen to keep their children at home, or have special circumstances to consider,” she said.

“However, Catholic schools will remain open and operational until we have been advised otherwise.”

Although the Australian Health Protection Principal Committee has recommended that schools remain open, Labor is concerned that the government appears to be keeping them open, in part, to ensure essential service workers like doctors and nurses can continue to go to work.

The shadow health minister, Chris Bowen, said the government should explain “what the trigger point might be for school closures” and “prepare now for health workers”.

“Let’s not use the fact that health workers who have their kids in school as a reason not to close schools if that becomes the right decision,” he told reporters in Sydney.

“Let’s plan now to provide the support for health workers.

“In Norway, the schools are closed but there is a skeleton teaching staff left in the schools for health workers, the kids of health workers so they can still go to work.”

In her letter, Plibersek called for “support for parents who are unable to attend work in the event of extended closures”.

“In particular, ensuring arrangements are in place so that health and other essential services workers don’t have to leave work and grandparents and other older relatives aren’t put at risk providing child care.”

Plibersek also made a range of demands including:

Ensuring all schools have access to adequate sanitation facilities.

Making sure online learning options are available.

Support for at-risk children and families.

Extra measures to ensure the health and safety of teachers and other school staff.

Special consideration arrangements to make sure school closures don’t unfairly affect a student’s marks, especially in Year 12.

“I have been contacted by a number of parents concerned that lack of soap or spring-loaded taps in bathrooms at their children’s school undermines good hygiene.”

On Wednesday the Australian Medical Association’s Western Australian president, Andrew Miller – who has previously called for the phased closure of all schools – refined his proposal to an “essential school attendance only” model.

“We have [proposed] a reduced school model,” Miller told ABC TV.

“Kids that need to be there for educational reasons or for family reasons, the children of critical workers or people who need to stay in the workplace and have no one else to help, that’s the sort of thing that would still keep going to school.”

The Victorian premier, Daniel Andrews, said that taking all students out of school “would do more harm than good” but “there will be a time when schools will be significantly disrupted”.

“In many respects, the best place for the kids at the moment is at school,” he said.

The Victorian medical officer, Brett Sutton, defended independent schools making “operational decisions” to close as a result of students withdrawing, while Andrews agreed it was “entirely a matter for private schools” whether to close.

“They need to make a call on the basis of the individual circumstances,” Sutton said. “I won’t stand in their way if that is an operational reality for them.”

On Wednesday the NSW premier, Glayds Berejiklian, rebuked independent schools which have chosen to shut down.

“We would prefer that everybody is absolutely on the same page when it comes to all the stakeholders in education,” she said. “There is no rationale for closing down schools.”