I can only talk about my experience. I have no idea about the NHS except a few threads where some of the issues raised have horrified me.
I came from Spain where we have a good health care system and I prefer Australia. We pay the top tax bracket for Medicare and have private health insurance but frankly, we've never used it.
Our GP bulk bills, which means they get paid the Medicare rate and don't charge extra. Most of the GPs I know in Sydney just bulk bills. My doctor is excellent but I can always change to any other doctor of my choice, There are no constrictions. Tests are free, from blood work, MRI, ultrasounds, x-rays etc, they take no longer than a week to be done and as I say, free of charge.
Specialists are private ones, referred by the GP. Most of them charge a fee and then you get the Medicare reimbursement. As an example my youngest son has an immune condition and sees his specialist a couple of times a year. With private health insurance the specialist is free but without it, we would be $70 out of pocket per visit, which is very reasonable. As Sydney is now in lockdown and my son needed some blood tests, instead of going to the Pathology service, it came to us. Arranged a date and time, arrived with a mask, took bloods and the specialist gave me a call a week later with the results. This on Medicare, not private health insurance.
Same son broke his arm last May. We debated going to the private hospital or the public and because the public was closer and my son was in pain, off we went. We arrived at Emergency (your A&E) at 10.00am and by 12.00 we were home again. He had been seen by bone specialist, x-rayed and cast made. Again, public hospital.
Ambulances are not free but if you are a pensioner or disabled, they are.
I find the system really good. It's a mixture of public and private that works really well.