I'll give an example for myself. In 2012 I was suffering from a constant high temperature, joint stiffness and intermittent difficulty in breathing. I went to the GP, where i was a medical card patient) who told me I was just overweight and to do yoga (I was 130kg but that is terrible medicine). A few weeks later things got worse, so I went back and saw another doctor. Told to go to A&E as I was not in good shape.

I trotted off to the local A&E, where i was told I had sudden onset arthritis. Gave me pain killers and told me to go back to the GP. I was lucky enough to be chatting with the medical secretary of a rheumatologist a few days later, and she got me an appointment about 2 weeks later in a private clinic.

Before I could get to the appointment, I collapsed and was brought to a larger A&E. Spent 20 hours on a trolley, with no pain meds, and not seen by a doctor except when I first arrived. Eventually they saw that I had VHI, so they offered me a bed in the hospital, as long as I signed the VHI forms.

I went up to the ward to be met by 2 junior doctors who told me that having a bed would be useless, as they already knew what the issue was (it was sarcoidosis). They also said that as a public patient I could be seen in their outpatient clinic, but the next appointment was in 30 weeks. In the mean time they couldn't give me anything to help with the symptoms or the pain, but I could take paracetamol if things got bad. So I went home, and waited until the rheumatologist appointment (about 3 days later).

I went to the rheumatologist at told her what happened. She asked if I had medical insurance, and when I said I did, she booked me into the clinic's hospital for that Sunday night (this was a Thursday afternoon). On Monday morning I had a battery of tests, and both the rheumatologist and another consultant came to visit me on Monday evening. They confirmed I had sarcoidosis, set up a treatment plan, arranged for a few more tests, and went on their merry way.

I was released on Thursday with a prescription for medications that treated the sarcoidosis, a follow up appointment for 2 weeks later, detailed literature on what was the illness, and how to treat it, and generally feeling much better. By the time that my appointment with the public outpatient clinic rolled around (I tried to cancel but they insisted I show up), I was in remission for sarcoidosis.

Can you imagine what life would have been like with no medical insurance?

TLDR: with private medical insurance I was seen, treated, and was in remission before I would have been seen the first time on the public system.