It's not as simple as "give up some of their personal wealth to pay more for an NHS they've already paid into". If you have 20K in your bank account and no property, you have to pay for your care. If you own 2 flats worth 1M each and have 10K in your bank account, you pay nothing. The so-called "dementia tax" would have make this situation fairer. Besides, they wouldn't have to release equity while alive, but the cost of care would be paid by the heirs upon selling or inheriting their parent's property.
Imagine a scenario where you have 20K in your bank account. You child inherits only 10K, because you had to pay for your care. Now imagine you have 2 flats and 10K in your bank account. Your child gets the whole lot. Does this look fair?
Theresa May, that, as far as I'm concerned, should be sent to the Hague for architecting the hostile environment for foreigners, had only one non-demented idea in her entire political career: making social care fairer, and that's what killed her premiership. I can't think of a more emblematic example of what is going on in the UK.