That's the fundamental point of insurance. You pay not to worry and to be taken care of if need be.
The point is that usually with public healthcare you still have to worry, because if something happens to you you'll most probably have to pay for them to take care of you anyway.
Well, public healthcare is good in theory, but it almost never works as it should. We might have one of those worse examples here in Poland but you can only count on it if you're relatively healthy. You can always see your doctor and waste his time with bullshit, but if you're ill, they will not treat you immediately unless your life is in danger 'here and now'. Cancer? Your problem, wait for 2 more years and fuck off. Because you don't pay them directly they will not hesitate to tell you that literally.
Also, you paid 20% of your salary every month for your healthcare but now you're over 80 years old and need a pacemaker ? Fuck you, you will not get it because you're too old and it's not worth "public spendings".
The government organization responsible for managing the system is one of our biggest debt creators. They have hundreds of people responsible for counting whether you're worth a certain surgery taking into account your age, other health issues and your life expectancy only. Bums aren't treated any different than people who are way more valuable.
In theory you can count to see a dentist but they will only fix your tooth if it's about frontal teeth, and will not treat you with an implant if you have any tooth remaining. Also, they use the cheapest materials that you wouldn't want in yourself and treat you like shit in the process, because you're being served "for free". If you want better materials you have to pay more than at a private clinic.
I know for a fact that it's not only like this here. I suppose that in America you guys might be used to great healthcare, because shitloads of cash go into that there. We use Canada as the prime example of what healthcare should look like too, and my aunt from UK was scared to get sick here but most countries with public health care suffer from the same issues (often worse). Private healthcare quality is almost always better. Public is almost always mismanaged and doesn't give a shit about people, because it's not paid for directly.