It helps the NHS in as much as it reduces load on them. Though what percentage of people would have to do that to have an appreciable benefit is another matter, and it doesn’t do anything about A&E.

But the more people go private the more excuse it gives the government to privatise the NHS ("it isn't working, look how many people are going private"). Also many of the private doctors are also NHS consultants. Getting more private work might just mean they reduce the hours they do for the NHS. So it isn't clear to me that going private helps the NHS, possibly the opposite. Also, if you pay your taxes, you shouldn't have to go private.

It seems to me the main factors here are:

a) Healthcare is getting more expensive due to an ageing population and ever more elaborate and expensive treatments.

b) The greed of the wealthy (who would rather have a third card than pay taxes to fund the NHS). Who also control much of the UK media.

c) The incompetence and venality of the UK government. How many hospitals could have been built by the money paid out in fraudulent COVID loans or trousered by mates of the government for COVID PPE?

d) The lack of long-term thinking and investment. We seem to change ministers like most people change theri underwear. It is beyond shameful that the NHS is hovering up doctors and nurses trained in developing countries to make up the shortfall.

The overlap of people who can go private, and use the NHS is very slim. It really would have an miniscule effect.

All complications from private healthcare spill into the NHS. There's no such thing, yet, as private A&E for example.

Healthcare is a life or death situation. I'm not going to abandon my own health because I think that if I don't do so I might hurt the NHS

I completely understand. Overall it would be best for everyone if we had a really well run and funded NHS. But it is better for the individual (who can afford it) to pay and skip the queue. It is basically 'the tragedy of the commons'.