As it applies to schools, you would need to reverse academisation and put the schools back under local authority control. Doing that alone would put big amounts of money back into the public service without costing any extra; you could divert it into recruiting more classroom support, facility refurbishments and so on.
It looks like a similar story for care homes but I don't know that sector particularly well.
From my experience care homes suffer from two issues directly to related to them not being publicly run, on top of the well documented issues with poor pay and unfilled roles.
The first is that astronomical fees are paid to enable an individual or company at the top to take out huge amounts of money.
The second is that as they view it purely in terms of how much money can be made there is a lot of interest in running care homes at the lower end of the needs scale, essentially old people who just want to live in a community and have a basic level of care assistance, but much less interest in running facilities for people with more advanced conditions and complex care needs. Essentially they want they easy money.
There's another issue, which you could in part put down to privatisation, and that's duplication of work. The amount of meetings, and amount of paperwork, I have that are just going over the same thing. That's because you have meetings with the care home itself, the NHS, the council all going over the same thing. The meetings with the NHS are largely because they have no direct control over the medical care being given, meetings with the council because they are paying (some) of the bill, meetings with the care home so they can try and upsell you extras.
I queried why there are so many meetings and was told by the council that it was common for care companies to keep billing them when people had either died or moved elsewhere.