[majewsky]

I'm a German with statutory healthcare.

> most [specialists] will refuse you on account of them being "fully booked"

I don't deny that this does happen, but it's not as inevitable as you make it sound. I've seen about a dozen separate specialists over the years for various reasons, and have never had such problems. There were a few outliers where I had to wait several weeks for an appointment, but that was only for non-urgent matters and I was never refused service. In most cases, I can get an appointment within 1-5 days of calling the doctor's office.

My suspicion is that such overload as you describe is a regional issue, so if, as an immigrant, you need to rely on certain specialists because of chronic ailments, it may be worth to investigate the availability of the relevant specialist doctor beforehand.