It's a lottery who will end up needing to go into a care home and who won't, and I don't think it's fair that it's self-funded, given how eyewateringly expensive it is.
NHS care is free at the point of need (in theory - the downside being you often have to wait). We all pay for that out of taxes and recognise, if we're sensible, that we're often going to be paying for other people's care, not our own, but few of us grudge that because it could be our turn next. Why is it different for social care?
I'd be happy to pay more in taxes or a one off insurance payment to guarantee that free, good quality care would be provided in my own home as long as possible and if/when not possible, in residential care. If in the end I didn't need that care and had therefore paid more than I needed, fine by me. Just like paying for any insurance that in the end you don't need. You're paying for peace of mind.
One way of funding social care could be from inheritance tax. Currently a married couple can leave up to £1m to their children without a penny of inheritance tax being payable. That's a huge amount of money. It would do a lot to reduce the inequality in our society if some of it went back to fund public services.
We should also be funding end of life care in hospices out of taxes. It's absurd that they're dependent on donations and charity funding to cover their costs.