I'm conflicted because of my own experiences.
DH ended up going private because despite presenting with the same problem 5 years in a row, the GP thought pain killers were more of a solution than investigation and proper treatment. In the end nothing would have changed the outcome; his problem is degenerative, but with proper treatment and understanding he may not have deteriorated so quickly. It may have helped him hang on to his job for longer, or could have prepared him for a job change. As it turned out his job was whipped from under him which aside from all the practical problems, gave him mental health issues too.
DD broke her leg, we drove her to A&E and had an X-ray and a temp cast on, referral for fracture clinic in the morning OMG and we were back in the car within an hour and a half.
Obviously I have lots of in between experiences, but I think it can and does work well, but not always.
One of the biggest issues I found when DM was in recently was lack of communication. It was incredibly hard to speak to a doctor and when I asked the nurse all she ever said was I needed to speak to the doctor. Doc came at 9am then 3pm on that ward so I was never there due to school runs an hour away, they also came round at 9pm when i wasn't allowed to be there!
I have had 2 DC referred to CAMHS for autism assessment. One had presented with signs from a very young age but was sent to the paediatrician who 'kept an eye on things' for a time before sending the referral forward. DC was 9 before being diagnosed. The other was referred by the GP straight to CAMHS when they were 6. He was also 9 when he got diagnosed. That's not good enough. People are struggling and fighting and needing help but not even getting a diagnosis for years. It's a failing.
Mental health services are stretched to bursting point. People are suffering.
I do think a lot of it is down to money. It's all very well saying they pay too much for paper clips, but the bigger picture says even with no paper clips, money is short.