From a macro-economic point of view, the King's Fund is wrong.
The government can quite literally create money. It can do that at the click of a button by the Bank of England, as it has done with the current crisis. It can also issue bonds. People are queuing up to buy bonds because the government is seen as a safe investment. Pension funds rely on deficit spending by the government because they own hold huge stocks of government bonds. If nobody wanted the money pension funds hold, they'd be in real trouble. While interest rates are so low, it really doesn't matter whether the government has a debt or its balance sheet is in deficit.
Most money the government "creates" or borrows eventually comes back to the Treasury via taxation, whether as direct or indirect taxation.
So why does a government need to tax?
There are many reasons. Firstly, it needs to control the money supply; otherwise there would be rampant inflation. Secondly, it needs to direct spending. It needs to collect the money back in a cyclical process. If it didn't (and left the economy to market forces) it would never be able to control the money going to health, education, infrastructure, etc. It would never be able to achieve its aims.
Thirdly, it seeks to redistribute wealth. People are selfish and, left entirely to market forces, those with more than average would never contribute to those with less than average. However, societies are more complicated than that.
Even the most heartless need the majority of people to have money they can spend. Otherwise, they wouldn't be able to make a profit. A developed country also needs educated and healthy citizens to be efficient workers. It needs good infrastructure for businesses to flourish. There's a very good reason that businesses (particularly those selling "intellectual" services) choose to do business in Europe, the US, etc rather than in most parts of Africa.
Anyway, I'm digressing.
The King's Fund is correct in claiming that the NHS is funded from taxation and NICs - on the government's balance sheet. However, the government could choose to create more money for the NHS, if it wanted. Eventually, it would need to collect the money back to avoid inflation, but any Conservative government is reluctant to collect more back from the wealthiest.
Extra money in the economy would almost inevitably find its way into the pockets of the wealthiest, but those people don't want to pay it back. That's the problem! They claim they've earned it by hard work and entrepreneurship, but that's not really the case in a marketised economy. The people making money out of the NHS are the drug companies, the equipment suppliers, the builders, etc etc. Increasingly, people providing clinical services to the NHS are making money. It's estimated that the NHS's own internal market (created by the 2011 Act) costs £30 billion in admin costs, such as negotiating procurement and legal fees for drawing up contracts.
Clicking a button to create money is the easy bit (as Sunak has found out) but deciding how that money is going to achieve its goals and claiming it back, so it recycles, is much harder.
(Sorry for the length of that - I've actually left loads out, but it's really not as simple as people seem to think.)