I don't think I have ever heard that claim before. I am American, had very good insurance through my employers when I was living in the US, and routinely experienced mediocre care. But I didn't have anything to compare it to at the time. After I moved overseas and experienced excellent healthcare I know better.
"Oh, you need a specialist. You will get a letter for an appointment in about 4 months." 4 months later when you go see the specialist, he decides he needs an MRI to diagnose. Wait another 3 months for the scan.
Unless you are literally dying right now, this is the reality of many health care systems. This scenario is about 3x more common in national health plan systems than in insurance based ones.
In other words, if you are in an affluent area with a well-funded NHS trust, you will receive prompt and plentiful care whenever you need it and for whatever reason you need it. Less affluent areas must prioritise to meet their budget constraints, so they will indeed make you wait for a long time, unless there is an emergency.
So, reports in the (non-UK) media that claim that it may take "4 months" to be seen by GP or a specialist doctor unless you're dying, etc, are a) exaggerated and b) don't take into account the regional differences in service, which can be very pronounced.