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He said with an ageing population and all the pressures on the NHS he could not understand why it was even being discussed.
He also called for a disparity in funding between Edinburgh and Glasgow to be corrected so the Capital could tackle its "horrific" problem of homelessness.
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Speaking to the Evening News after a visit to Drylaw neighbourhood centre, Mr Sarwar said May's council elections here would be about getting investment back into the Capital and showing Edinburgh was still a thriving international city.
And he repeated his hope there would be "no formal coalitions" after the election and instead cross-party co-operation on individual issues.
The controversial plan to close five council-run Edinburgh care homes was announced in June as part of a review of care provision in the Capital.
One closure, Drumbrae, has gone ahead, with the home switching to an NHS-led complex care assessment unit.
Edinburgh's Labour group has come out against shutting the other four – Clovenstone, Ford's Road, Jewel House and Ferrylee.
And unions are campaigning strongly to save the homes and maintain public-sector care home provision.
The body responsible, the Edinburgh Integration Joint Board (IJB) – made up of council and health board representatives – postponed any decision until after the elections.
But Mr Sawar said: "For anyone to suggest we should be shutting down care homes is just unthinkable, particularly when you think about the ageing population, the huge challenges we've seen during the pandemic around the care sector and the huge challenges in restarting our NHS and having a functioning care system alongside it.
"What we should be talking about is how we have greater investment, how we improve the facilities and the level of care, how we support, attract and retain the workforce and that we pay the workforce properly.
"The idea we would think about closing care homes baffles me. I can't even understand why there's a discussion.
"There's a debate to be had about whether they need to be upgraded, redesigned or reconfigured, but the debate cannot be about shutting down care homes."
He was also unequivocal on the need for Edinburgh to get more government cash to tackle homelessness. The city council has lost out on £9.3m funding because unlike Glasgow it has not transferred the responsibility for the issue to the IJB. Despite council lobbying, the Scottish Government has shown no sign of flexibility on the matter.
Mr Sarwar accused Housing Minister Shona Robison of trying to turn Edinburgh's legitimate plea into an issue about "structures, models and bureaucracy".
He said: "The harsh reality is there is a disproportionate number of people who are homeless in Edinburgh compared to other parts of Scotland.
"One person homeless is one person too many. and the scale of the numbers in Edinburgh is horrific. Edinburgh should get its fair share of funding to eradicate homelessness in the city and play its part in what is a fundamental human right about being able to live in a safe secure home.
"The funding disparity must be corrected. Politicians and bureaucrats can argue about systems. People just want it sorted."
With the council elections looming in May, Mr Sarwar said the crucial issue was getting investment back into Edinburgh, restarting the tourism industry and boosting the local economy.
"That's what Labour is going to be campaigning on – how to support the local economy and local quality public services and support community organisations across the city as well, to demonstrate that Edinburgh continues to be that thriving global city that frankly has been let down I think by a Scottish Government that continues to see its budget go up but thinks loal authorities are an easy cut to make."
Mr Sarwar has made it clear he does not want his party to do deals with either the SNP or the Tories to get into power in local government.
He said: "There's a fundamental opportunity to change the way we do local democracy.
"The SNP have ruled out coalition with the Tories and are challenging Labour to do the same; the Tories have ruled out coalition with the SNP and challenged Labour to do likewise.
"Rather than that being a negative let's turn it into a positive. Let's have no formal coalitions and instead maximise the number of Lab councillors and Labour councils and then strengthen local democracy in that people will have to work together on individual issues, based on what's right for local communities. That's the way to do local democracy rather than party political stitch-ups."
Mr Sarwar said he had been glad of the opportunity to meet local people during his visit to Drylaw neighbourhood centre.
“It’s a lovely community and I was hearing about the work the community centre does and the ongoing challenges they're facing and hearing directly from the service users about the issues of isolation and loneliness – issues that were obviously huge during the pandemic but are also huge coming out of the pandemic, particularly for those who are older.”
He said community centres were crucial to people’s wellbeing. “The ability for people to get out to meet their friends, make new friends, share joy and sorrow with others, helps to protect their mental health and keep them away from GP practices and mental health services which are already under pressure and helps to give them a quality of life.”