Sounds like you are living in Germany. I would consider it public health insurance but is a long way from universal healthcare. Although you may not be paying directly for a treatment, the whole medicine sector is private (I.e. there are no doctors directly working for the insurance provider). That causes some inefficiencies like hours of waiting in ER, doctors not accepting new patients etc.
Compared to other systems in the EU (where doctors work for the insurance provider, most often times the government itself) processes are way more streamlined, like getting a GP assigned depending on where you live. But “you get the healthcare that you need, not the one you want”. Other issue is that they are almost running at capacity, so things like a pandemic tip the system over quite easily…