Of course, one problem with the current bureaucratic system in the Nordic countries, is that some people slip through. If for example you start a personal company, and later fail, you won't get unemployment benefits (unless your company was a corporation and you hired yourself, of course). At least, that is the case in Norway.
That's the case in many European countries with sophisticated welfare systems. In Austria, we can pay extra (I think ~1.5% of income) as self-employed to get unemployment benefits but I don't know how popular that option is in practice.
Personally (as a highly employable, fairly senior software developer, so I'm unlikely to be out of a job as long as I can work) what's worse than lack of unemployment benefit is dealing with times of sickness. I suffered from Lyme disease for about 2 1/2 years until it was diagnosed properly and treated, and in addition to the reduced income due to being unable to work at times, and the extra cost of healthcare (despite the insurance it's not exactly free) in Austria, they still expect you to pre-pay taxes and pension & health insurance contributions based on your earnings 1-2 years ago. In theory you can claim some money for a few weeks if you get sick, but (a) you have to completely shut down any business activity for that time and (b) you have to be declared unable to work by a doctor. (Unlikely, considering the doctors were all telling me there was nothing wrong with me and it was "stress"…)
A basic income would at least help deal with the reduced income due to sickness & extra costs due to healthcare. The pre-payment aspect is probably peculiar to Austria and apparently frequently bankrupts profitable small businesses.
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.
I find your question a bit unclear, but I'll try to answer.
Even in a best-case scenario where the politicians who are creating the laws/rules are well-intentioned and really are trying their best, I guess it's impossible to think of every single real world scenario. Thus, there will always be people who will fall through a complicated safety net.
As for why entrepreneurs aren't always covered in Norway? I'm just speculating here, but anyway: Since WW2, Norwegian politics has been dominated by The Labor Party (Arbeiderpartiet). They usually look out for ordinary workers, which tend to have it pretty good in Norway. Others groups, such as entrepreneurs, drug (ab)users, farmers etc. aren't always treated so well.
