Arguments about Germany's energy independence would be far more compelling if they were not, for example, currently building giant pipelines to import Russian natural gas.
That's mostly for heating, though. Heat pumps are being increasingly used in new homes but nearly all homes older than 10 years are using gas or oil furnaces. Electricity prices are also very high, amongst the highest in the world, so heat pumps aren't as cheap to run as in other countries. It's definitely a problem that needs to be fixed, potentially within the next 30 years because that's the timeframe where the law requires old furnaces to be replaced.