In Germany (which has socialized healthcare) I've heard of plenty of entrepreneurs and freelancers who aren't on health insurance because they can't afford it. What's worse: if they eventually do sign up they are legally required to pay fees for the entire time they didn't have health insurance. Good luck paying the backlog if you've been off healthcare for a few years.
If you're self-employed, a lot of the welfare system doesn't really apply to you either. Heck, if you are self-employed (or running your own company) and get pregnant, you don't even get the benefits an employee would (you get child benefits because those are for the child not you but that's pretty much it).
You get those benefits if your employer pays you for them. The problem is that you, as the employer, have chosen not to pay you, as the employee, those benefits. That money has to come from somewhere and if the employer doesn't pay it, the employee doesn't get it.
Unless I'm misinformed, you're actually wrong. As far as I was told, you can pay into the social welfare system but as you don't qualify for certain benefits the only thing you can do is get a refund of the most recent payments (not all of them, mind you).
So you're of course free to pay into the system, but as you're neither required to pay nor eligible for the benefits, it's mostly a waste of money.
In the case of parental leave, the employer only pays as much as 50%. The rest is paid by the tax payer (or the government agency, rather). Plus, of course, the employer has no say in whether they pay this or not. They're required by law -- thanks to the arm's length principle ("Fremdvergleich") it wouldn't matter whether you as associate ("Gesellschafter") pay for yourself as the employee; at least if you're incorporated, not merely a civil law partnership ("GbR") or DBA ("Kaufmann").
Of course the situation is entirely different when it comes to healthcare: if you're an employee of your own company (i.e. you incorporated, otherwise you are the company) it's considerably more difficult to opt out of healthcare entirely. The healthcare-free entrepreneurs I was talking about are regular DBAs or freelancers and the only reason they don't have healthcare coverage is that they can't afford it (which of course technically means they should actually be insolvent as a business but it's difficult to draw the line between trying to survive as a business over a rough patch and desperately holding on to a sinking ship).
Oh, and another fun excerpt of German law: senior staff (which includes general managers, i.e. you if you incorporate and employ yourself) is exempt from a fair share of labour laws. On the plus side this means you don't have to obey vacation rules for yourself, but I'm sure you can also think of some of the negative implications.
If you're self-employed, a lot of the welfare system doesn't really apply to you either. Heck, if you are self-employed (or running your own company) and get pregnant, you don't even get the benefits an employee would (you get child benefits because those are for the child not you but that's pretty much it).