Older people should receive free help to eat, wash and get dressed in a move which would improve their health but need to be funded by a 2p tax rise, a thinktank has said.

The proposal, by the left-of-centre Institute for Public Policy Research (IPPR), highlights the growing political consensus that personal care should become free for over-65s. If implemented, it would bring England into line with Scotland, where such care has been free since 2002.

The IPPR argues that the key principle underlying access to the NHS – free care at the point of need – should be extended to this element of social care services in England.

Doing so would remove what critics say is a deeply unfair system in which more and more people of pensionable age are having to use their savings to pay for care received at home that is vital to their independence.

The switch would cost an extra £8bn a year by 2030 but could be paid for by raising income tax by 2p or National Insurance by 1.3p, according to calculations in a new IPPR report.

The NHS would save £4.5bn a year by 2030 because older people would be in better health as a result of improved support at home and so would end up in hospital less, it says. Cuts to local council budgets since 2010 have contributed to hospitals becoming routinely full all year round.

The NHS’s bill for providing “continuing healthcare” to those with high-level medical needs would fall by £3.3bn, fewer hospital admissions would save £270m and improved end-of-life care in people’s homes would yield a further £267m saving. A fall in the number of patients who remain in a hospital bed despite being fit to leave would free up another £670m.

The report comes days after Jacob Rees-Mogg, a prominent rightwinger and leader of the European Research Group of pro-Brexit Conservative MPs, said he supported the principle of social care becoming free and paid for by government, not individuals.

“It is far better to pool risk and for the taxpayer, where appropriate, to step in and help those who would face ruinous costs on their own, making social care largely free at the point of use. This is something we can afford as a nation, if we can get our priorities right,” he said.

However, those receiving care should also pay £5,000 a year to help cover the costs, he added.

Last month, Conservative MP Damian Green said social care should be modelled on the state pension in a paper for the Centre for Policy Studies, a free market thinktank.

“In 1948, politicians were brave in making the NHS free at the point of need and funded out of general taxation. We need our politicians today to be just as courageous and do the same for social care,” said David Behan, the chair of Health Education England, the NHS medical training agency.

“The hallmark of a civilised society is how well we treat the most vulnerable, including our elderly parents and grandparents. At the moment, we are failing them, but it doesn’t have to be that way.”

Labour has pledged to introduce a National Care Service and massively expand access to free social care if elected.

Barbara Keeley, the shadow minister for social care, declined to say if she backed the IPPR’s call for free personal care. She and the shadow chancellor, John McDonnell, have been discussing whether to include such a pledge in Labour’s next manifesto.

“At the last election, Labour pledged to set up a National Care Service, cap care costs and spend £8bn more over the parliament, and last month we announced plans to ensure that 160,000 more older people receive help at home with their care needs,” said Keeley.

A government green paper on reforming social care has been delayed many times since its intended publication last year. It is expected to include proposals on “risk sharing” to cover the cost of expanded social care support and stop retired people having to sell their home to pay for such help.

A Department of Health and Social Care spokesperson said: “We are committed to ensuring that everyone has access to the care and support they need. We will set out our plans to reform the social care system at the earliest opportunity to ensure it is sustainable for the future.”