The public insurers are not private organizations. The reason they are called "gesetzlich" is exactly that: they exist by law. You are right in that they are no

The public insurers are not private organizations. The reason they are called "gesetzlich" is exactly that: they exist by law. You are right in that they are not run by the government (be it state or federal) but are their own entity. They are however self-governed and currently hold elections for their board that undergo the same scrutiny as any other public election. They are public entities in full right. Not paying your fees has the same consequences as not paying taxes and they have more far-reaching powers than a private organization ever could.