My experience with one of the larger HMO organizations here was very disconcerting.
I was bedridden with spinal joint pain for several days. They were unable to see me in a reasonable timeframe (2 week wait minimum), which meant by the time I was able to come in for a blood test the acute inflammation I was experiencing had passed, preventing screening for the condition I was concerned about (my dad and brother both have ankylosing spondylitis, making it extremely likely I do as well).
Worse than that, I found I had to educate the RN that was responsible for interpreting my test results (specifically that ankylosing spondylitis shows negative rheumatoid factor).
In addition to this, I've watched what's happened to friends with serious but not readily diagnosed diseases. One was in advanced stages of liver failure. In the end she only got a diagnosis by going to a hospital, making herself a bother and refusing to leave until someone took responsibility for actually finding out what was going on. The result was she avoided an unnecessary liver transplant. Her diagnosis was an uncommon but easily treatable autoimmune condition.
There's roughly 50% odds the transplant would have killed her by now (this was some 10 years ago). Taking an adversarial approach and having the support of a friend's mom who worked for the business side of the HMO in doing this probably saved her life.
It shouldn't have to be this way.
It's abusive to doctors that things are set up this way as well.
In any case, I don't mean to rant, but I just wanted to provide some counter anecdotes to to your feeling that the callousness and incompetence of medicine in the states was being exaggerated.
My stepfather went for a routine visit to the doctor in Vienna, Austria. They found a suspicious object, and decided to treat him immediately, and he was operated on the very next day. It ended up being benign, but I was amazed at their speed of treatment.
My uncle went for a cardiologist evaluation here in Montevideo, Uruguay (where we have a form of socialist-style medicine in the style called Mutualism). They found a suspicious spike in his heartbeat, which they suspected to be a treatable syndrome (Wolff-Parkinson-White), and decided to do some special evaluations. He was treated (with full anaesthetics) one week later (at almost zero cost).
Both were quite good experiences, times are a little slower in Uruguay but everybody has access to basic medicine.
That said, I suspect that for difficult-to-catch diseases, you have to make yourself a bother, otherwise you might slip through the cracks like your friend almost did (doctors here in Uruguay are heavily penalized if they take too much time with one patient).
[Note: I realize it might just be that my insurance is ridiculous. I'm looking into alternatives.]