There is a lot wrong with the health system here, but:

. We tried in vain to make an appointment at any local doctor's office but there were no appointments for that day.

Usually there should be a Hausarzt nearby and you just show up and get this notice. There is no need for an appointment (to be fair COVID changed that in some cases)

It just feels like this German healthcare miracle

Whoever claims there is a german healthcare miracle is misleading you, we really don't claim that here.

and start paying into social medicine.

This is how systems like this work: everyone pays insurance and then the system/insurance pays for you.