Why having the government in charge of something means it's going to be inefficient?
Ultimately the government needs to be in charge of healthcare because healthcare is not a simple service but a human right. In many ways, we might trade the benefits of competition (which in the long run do end up ruining the system, just look at the USA) for the benefits of democratic ownership.
In any case, as others have pointed out, the system can be turned inefficient if government by government you dismantle the system a little bit and that's precisely what's been happening in the UK. However, this is not something that has happened in other countries in Europe and in general socialised healthcare gives us the highest life expectancy numbers.