No, Winchester 73, a primer is not always enough to push the bullet clear of the barrel.
The answer of what to do with any failure to fire depends on the circumstances, and it is not a good idea to have some automatic response.
If you are on a range, and suspect a hang fire, keep the gun pointed down range for at least 30 seconds, then keep the gun away from your face while you open the slide/cylinder (if you can) and determine the problem.
If you are in a self defense situation, then you have to take a risk; if you have a revolver, squeeze off the next round. If you have an auto pistol, rack the slide and squeeze off. And hope that anything bad happens to the other guy.
Some of the "super training" guys believe anyone with a gun in hand should always act automatically, without ever thinking. I can tell them that the guy who doesn't think can be in big trouble, one way or another.
Jim