I hope no-one minds me posting here for a first post. I've just read through several dozen posts in this thread and the central concept running through it is erroneous.
The UK government funds healthcare and it is a sovereign currency issuer, so the money it spends on healthcare (or anything it spends on) is spent into existence, not 'borrowed' or 'recovered' from anywhere as 'funding'. You pay tax for a different reason which I'll get to in a moment. The NI system as conceived of is actually obsolete post 1971, I'll get to that as well. But first...
The government is mandated to provide healthcare spending, of this there is no dispute. Obviously since 1948 so-called 'fiscal conservatives' (as they refer to them in Uncle Sam-land) have always been anti-government spending on such large-scale social infrastructure. Until the abolition of the gold standard in '71 this had some merit; since there had to be a real consideration of issued money in relation to mobilisation of resources (and any resource limits). Post 1971 it is meaningless as a core concept. The question of resource limits always remains, the question of funding does not. The government can and does fund anything it wants to fund. That should be evident from at least 2008 onwards in the very public raising of the deficit ceiling after the financial crash. I say 'deficit', forget the word 'debt' because it is a red herring. The word deficit is more important, because it is gravely misunderstood.
Under Thatcher - itself under the sway of Friedman monetarism and the hare-brained concept of 'money-supply control', the idea was that government would aim to practically eliminate government spending and hand all mechanisms for goods and service provision to the private sector. They soon discovered that this doesn't work, but at the time still didn't know how or why (largely because all the people who actually understand monetary operations are few and lots had died off from the 1930s to the 1970s). Whilst knowing that of course private entities do actually provide goods and services, they seemed to be utterly confused about how such operations were funded.
Viz...the concepts of government currency issue as 'spending'; the purchase of services by government (a huge section of the economy, perhaps the largest and certainly the core of many more periphery activities); the concept of loan elasticity in banks: i.e. loans create deposits, not deposits 'fund' loans, which is fantasyland. So by attempting to curb their Thatcherite deficit (largely on social public spending) and 'control money supply', they simply throttled the supply of currency (credit) to people who needed it. See, when the government sector doesn't 'spend' the private sector is deprived of funding. This is what happens in a low-spend economy which also taxes. Since tax is what happens when government aims to collect and return money spent into the economy - it is destruction of spent money to create 'fiscal space', should this be required. The raising of tax by central government is not to 'fund' it is to create space for new spending (which is low in an economic ideology favouring tiny deficits). Now there are some people who like to avoid tax and find ways to do it. They are usually people who are meant to be paying larger amounts of tax on higher profits, but they avoid it. Since the monetarist governments are deficit-shy because they don't understand what it means, they drop that creation of fiscal space onto people who can't avoid tax. The fact that they actually don't need to do this to spend in a scenario of chronic underspending is evident. Chronic underspending where needed that is; not into the pockets of corporations who drain away government spending.
Now to the issue: who pays for social care? It is funding from government spending, end of. When it isn't, that is an ideological and perhaps moral choice, not a financially motivated decision. The small taxpayer should be furious that he/she is being hit for fiscal space because 1) government is obsessed with the monetarist notion of tiny deficits (equals: throttling of currency to private/domestic sector) and 2) that large tax avoiders are allowed to block up fiscal space with their unspent income largely derived from public spending (wholly derived from government issue). Since in the health sector the bulk of it is public spending going to people and organisations who are 'charged' with providing services, but in fact take the money and increase the prices, while delivering minimal return because they are profit-motivated.
The question dividing it into Mrs A who has a £300,000 house and Mrs B who lives in a council flat is a diversion from the central problem. Once you break the policy of universal coverage by means testing you break the entire thing. The problem is not there, it is in the false concept of government being currency-limited rather than the country having an (eventual) resource limit (yes Mrs May, you do have a 'magic money tree', use it wisely!). And the muddle-headed notion that therefore it is funded by private individual means to 'take up the slack'. The concept of privatisation in public service is a notion backwards in relation to monetary/fiscal operations. The private business sector (like you the private domestic sector) is a currency user, not an issuer, so they run out of it even when they don't run out of resources. They have, nevertheless, worked out a way to get a direct line of government credit, while the bulk of the population hasn't and has to rely on private sector crumbs.
You can have high-quality social care. If I mentioned something above, but didn't come back to expand upon it, please say so.