@Steven *********
I'm not saying there are never problems (depending on wrong insurance or other problems), but it is definitely possible to have direct billing if you have the right insurance.
In my view :
First of all, choose the right insurance.
Second : make sure you know the right steps and the correct communication channels before you need it, write it down on your insurance paper. Be prepared. So you don't lose time and avoid miscommunication problems at the moment you are too sick to deal with anything.
Third : get your insurance through a good local agent. When there is a problem, the help of your agent will maken a huge difference clearing up any mess.
Learned from experience.