Waiting times are not only for non-essential things. The only thing with no waiting time is A&E if you are in immediate danger of death.
The NHS is some "safety net" that does the bare minimum at this point, or no longer anything at all (eg. dentistry).
Interestingly, nowadays if you have a good job and private insurance in China you get better routine checks than in Europe/UK. Even Chinese hospitals are extremely thorough and quick compared to British ones (at least the main Chinese cities)
There's an inflection point, sure, where you start spending enough privately that the insurance is worth it, but so many of these things are so cheap here anyway that the gap from between where it starts to pay off until you're ill enough to get reasonably fast treatment with the NHS is quite narrow.
Sometimes people think it is wider than it really is because people want and push for treatments that are not medically indicated. E.g. patients pushing for pointless MRIs is common enough that same-day MRIs is a huge industry here even though they only make an impact on outcomes for very specific symptoms.
For instance the UK have very bad outcomes for cancer because things tend to be caught late and treatment delayed thereafter. Frankly, for anything potentially serious or time-sensitive I would go straight private (and in fact the NHS tells you to do that when they ask you if you have private insurance).
The NHS has much bigger problems than funding. Even the new, left-wing government has indicated that they won't increase funding without reforms.
It's always very convenient to starve something of funding for years when you want to insist something needs to be reformed before you can spend as much as it costs to provide service at an adequate quality.
Reform or no reform, outcomes won't improve without increasing the funding, as no other system in a comparable country manages to deliver more at the NHS cost level.
Another issue is that the NHS is a religion. It is blasphemous to suggest departing from free-of-charge delivery or private involvement (although that's already what we have). Even suggesting "reform" is badly received.
I'm from the US so excuse any oversimplifications, but over the past couple decades I've noticed a trend of US-ification in British politics. I would be careful with looking to greener pastures.
The French system is based on a mandatory health insurance, now with mandatory private health insurance in addition to that.
GPs and most health professionals are private practices that set up shop like a, say, lawyer would and there is big private sector involvement up to hospital level.
I think there are many similar examples throughout the world.
This is partly why I find this focus on the US system in the UK puzzling. My best guess is that it is used as a scarecrow by those opposed to any changes to the NHS.
Here's an NHS case.
Case 1. My friend's wife had her body going numb, like, completely senseless. They didn't have a private insurance back then. It was maddening hard to get past the GP. And then there were numerous tests, and queues, and... She had a viral brain infection that was, luckily, easy to fix. Could go horribly wrong should they wait for a couple more months.
The problem is that GPs are afraid to assign tests other than the most simple blood tests (unless it's an obviously broken bone).
Case 2. My wife fainted in the middle of a family dinner. It looked serious, and I recognized the sympthoms thanks to dr. ChatGPT. Our car was at service that day. I was trying to get an ambulance but calling all the publicly available NHS numbers but all we got were endless questioneers. We had to force the situation by taking a taxi right to the hospital door. She had an urgent surgery the same evening, quite serious.
Again, there's just TOO MUCH backpressure from NHS at the entrance. Once you get past these filters, it actually it decent.
OTOH, with my highest tier private insurance I had an MRI scans, kidney checks, blood tests all done within a week or two, mostly because of how I had to coordinate things between various private clinics. Not fast, sure, but acceptable.
As for tests, one of the reasons they "are afraid" has nothing to do with being afraid, but what is medically indicated. A lot of private services will do everything "just in case", the NHS won't. That means you often get people wondering why they've not been sent to an MRI for example (as one of the most common examples), because it's only actually affecting outcomes for a very small set of diagnoses. But some GPs certainly do get it wrong, and people need to be more aggressive about changing GPs if they feel they're not being heard.
So how it works, unfortunately, is that GPs get a lot of pressure to not do things "just in case". But "just in case" is the only way you can notice things that are wrong on a deeper level.
What, for example, is "back pain"? In 90% of the cases this can be fixed by, say, 10 sessions with a physio. But sometimes it can mean something serious, like a spinal disk injury.
Or sleepiness. 98% of that can be fixed with lifestyle and diet changes. But sometimes this can a sympthom of something scary.
Or my wife's example: didn't feel quite right for 3-4 weeks then fainted. Turned out to be serious. The full GP discussion would have taken weeks.
Or my friend's example: increasing numbness, nothing serious at first. Major brain infection. The GP actively tried to downplay things.
I don't know if there is a solution to this. Right now it feels that without chatgpt-assisted self-diagnoses and being aggressive with getting over the GP wall the system tries to avoid helping out.
I'm sure that if you're rich enough, you have fantastic care available in the US, but from what I've seen of prices for pretty basic stuff, I'm not so convinced most of the top 20% in the US have healthcare I'd consider tolerable.