What in Germany is called Steuer is not the same as taxes. Taxes are just the sum of your mandatory contributions from your gross income, including contributions for health care, pensions, public broadcasting, etc. Steuer is a part of your taxes.

The cost of “full insurance” is something like €500-1000, no one on Bürgergeld and few on Arbeitslosengeld are going to be able to afford that. Though you’re right that it doesn’t make any sense to have health care coverage tied to your employer when the insurance is mandatory for everyone anyway - that’s a bureaucratic relic from the past.

Health insurers for mandatory insurance in Germany are not private companies but semipublic nonprofits.