Examples:

If you get in a car accident, have a stroke, drug overdose or get cancer, you get almost free world class treatment in the public hospital urgently.

If you hurt your knee while playing tennis but it doesn't prevent you from working and living a normal life (but you can't play tennis with it), you go on the low priority surgery waiting list for public health care or get it faster from private sector via insurance or paying out of pocket.

Does this make sense?

The private health care is basically for skipping the line and most employers provide it for you so that you miss no time from work.

My point is that you are also paying taxes for that free healthcare while attending private.

My point is that you don't need the private health care, which also means they are not in a position to rob you if you choose to be their customer.

Even if you don't use the public health care it's there for everyone as a safety net in case something is not covered by your employer or insurance, you are between jobs etc.