I do have a solution.
If, when reaching 60, and before 67, a person has large assets - a home and savings valued at over £800k*, they then have the opportunity to do either
(a) continue as it is now and risk losing their home which they can't guarantee to pass on if they need expensive end-life care
or
(b) pay 10% of those assets** (so £80k if your assets are £800k*) into an insurance fund owned by the state. If you then need care, it's paid for***, done, and your relatives looking after you needn't worry. Of course anyone dying early, or living long and healthily, doesn't get the benefit from that insurance - in which case the money goes to the state to pay for the care of others.
*That figure would need a starting point, say £500k?
**Between 60 and 66, many obtaining a pension get a lump sum which could go to, or indeed cover, that cost
***The money could be given over the course of 6 years, maybe even encourage downsizing to free up cash to pay and also make larger property available for young families
The argument is, of course, where do you start, and would those below the assets threshold get free care? Naturally those above may well spend quickly to get below thresholds, but that can happen today to get to below £23k....however to do so deliberately risks being an offence.