Before you say just find a new job. I'm tired of looking for a new job because the insurance started getting worse at a company.
It doesn't matter which doctor I see it won't change what insurance decides.
Fuck private healthcare.
Your insurance company may sometimes refuse to pay for tests recommended by your doctor because neither you nor your doctor are directly paying for tests so the insurers are the only parties whose job is to push back on over-testing. The USA is famously considered an over-tested and over-medicalized society in general so arguably they could do a lot more of such pushbacks. If you do some research and think that in your case they're overshooting, then you need a different health plan. The fact that you can't get one due to the job tie is indeed a really broken aspect of US healthcare, and the fix is to fix the tax code so there's no benefit to having employers pay the premiums.
And that is why this tragedy happened. The outcomes here are terrible but exactly what you'd expect and what the NHS's critics have been saying for decades. Talk to people who used to live in the Soviet Union and you'll hear stories indistinguishable from this one. There are people and equipment there, in theory the ingredients for success are available, but nobody seems to care about the results. So the system steadily degrades into ever more dysfunctional states, the problems are obvious to everyone who interacts with the system and yet it cannot/will not improve.
This is exactly the reason communism failed and now the NHS is failing in the exact same way, for the exact same reasons. It's not even a new set of problems, it's just that the state of collapse is now so bad that even the most ideological people (e.g. Guardian editors) are starting to open up about it.
Sadly this poor woman is still a long way from being able to understand what happened to her family. Her conclusion at the end of the article is "don't blindly trust doctors, research things on the internet and speak out". Yeah, and then what? She was already doing those things and the doctors/nurses were ignoring her. Patients in the NHS are totally disempowered. It's not like a private sector system where you have options and could go to a competitor. If the doctors aren't diagnosing your child's problem properly because they're WFH then, well, you're just screwed: the child is going to die and the resulting investigation will be a whitewash in which the doctors who messed up the worst get promoted. That's what happens when you have one state run system for everyone.