Bingo. I live in Berlin. I know loads of folks who found Delaware corporations, use deel.com to work for them, and eventually found a GmbH when they need to.
Of course, the immigration question wouldn't matter for other immigration situations like holding German or EU citizenship, German permanent residence or EU long-term permanent residence, family reunification permits, or anyone who actually gets explicit approval to be self-employed. And the health insurance question wouldn't matter for people switching from employment (with public health insurance) to self-employment.
Practically though, the immigration office's bureaucrats are by definition as far removed from entrepreneurship as they can be, and might struggle to reconcile the documents you have with the list of documents they are told to ask for.
My impression is that working for a company one substantially owns or controls, regardless of where it is registered, can lead to them deciding that it's self-employment and disqualifying a third-country national from using that work as the basis for an EU Blue Card or GKV.
Also, the rules for obtaining an EU Blue Card (or a skilled worker permit) require the employer to have a place of business in Germany, and that might be hard to demonstrate when the only tie to Germany is the location of the applicant. No idea if just renting a German office address for the business would suffice. At least Americans, Canadians, and other §41 AufenthV Abs 1 nationalities can bypass this obstacle with the §26 BeschV permit instead, with some extra approval and priority check paperwork. But this doesn't help with the issues in the previous paragraph of potentially being viewed as self-employment for immigration and insurance purposes.
Sorry to be the one to break it to you but essentially they're all committing tax fraud.
They're all committing tax fraud and thinking they're in the clear, only getting away with it due to lack of international cooperation. It will bite them in the ass sooner or later, the taxman always gets his cut. The only way to avoid it is to physically move out.